COUPLED DYNAMIC ANALYSIS OF MULTIPLE UNIT FLOATING ... · rotor-floater-tether coupled dynamic...

COUPLED DYNAMIC ANALYSIS OF MULTIPLE UNIT

FLOATING OFFSHORE WIND TURBINE

A Dissertation

by

YOON HYEOK BAE

Submitted to the Office of Graduate Studies of Texas A&M University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

Approved by:

Chair of Committee, Moo-Hyun Kim Committee Members, John Niedzwecki Jun Zhang Alan Palazzolo Head of Department, John Niedzwecki

May 2013

Major Subject: Ocean Engineering

Copyright 2013Yoon Hyeok Bae

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ABSTRACT

In the present study, a numerical simulation tool has been developed for the

rotor-floater-tether coupled dynamic analysis of Multiple Unit Floating Offshore Wind

Turbine (MUFOWT) in the time domain including aero-blade-tower dynamics and

control, mooring dynamics and platform motion. In particular, the numerical tool

developed in this study is based on the single turbine analysis tool FAST, which was

developed by National Renewable Energy Laboratory (NREL). For linear or nonlinear

hydrodynamics of floating platform and generalized-coordinate-based FEM mooring

line dynamics, CHARM3D program, hull-riser-mooring coupled dynamics program

developed by Prof. M.H. Kim’s research group during the past two decades, is

incorporated. So, the entire dynamic behavior of floating offshore wind turbine can be

obtained by coupled FAST-CHARM3D in the time domain. During the coupling

procedure, FAST calculates all the dynamics and control of tower and wind turbine

including the platform itself, and CHARM3D feeds all the relevant forces on the

platform into FAST. Then FAST computes the whole dynamics of wind turbine using

the forces from CHARM3D and return the updated displacements and velocities of the

platform to CHARM3D.

To analyze the dynamics of MUFOWT, the coupled FAST-CHARM3D is

expanded more and re-designed. The global matrix that includes one floating platform

and a number of turbines is built at each time step of the simulation, and solved to obtain

the entire degrees of freedom of the system. The developed MUFOWT analysis tool is

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able to compute any type of floating platform with various kinds of horizontal axis wind

turbines (HAWT). Individual control of each turbine is also available and the different

structural properties of tower and blades can be applied. The coupled dynamic analysis

for the three-turbine MUFOWT and five-turbine MUFOWT are carried out and the

performances of each turbine and floating platform in normal operational condition are

assessed. To investigate the coupling effect between platform and each turbine, one

turbine failure event is simulated and checked. The analysis shows that some of the mal-

function of one turbine in MUFOWT may induce significant changes in the performance

of other turbines or floating platform. The present approach can directly be applied to the

development of the remote structural health monitoring system of MUFOWT in

detecting partial turbine failure by measuring tower or platform responses in the future.

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DEDICATION

This dissertation is dedicated, with love and respect, to my parents.

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ACKNOWLEDGEMENTS

First and foremost, I would like to warmly express sincere appreciation to my

advisor, Professor Moo-Hyun Kim, for his continuous support, invaluable advice, and

warm encouragement throughout the course of this research. His insight and comments

always helped me complete this thesis.

I would like to thank my committee members, Dr. Niedzwecki, Dr. Zhang, and

Dr. Palazzolo, for their guidance and support throughout the course of this research.

I also want to extend my gratitude to the American Bureau of Shipping (ABS)

for their technical and financial support and the National Renewable Energy Laboratory

(NREL) for their essential resources during my Ph.D. study. Thanks also go to my

friends and colleagues and the department faculty and staff for making my time at Texas

A&M University a great experience.

Finally, thanks to my parents, parents-in-law and my sister’s family for their love

and encouragement and to my wife J.Y. Kim, for her patience and love.

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TABLE OF CONTENTS

Page

ABSTRACT .................................................................................................................. ii

DEDICATION ............................................................................................................. iv

ACKNOWLEDGEMENTS .......................................................................................... v

TABLE OF CONTENTS ............................................................................................. vi

LIST OF FIGURES ...................................................................................................... ix

LIST OF TABLES ..................................................................................................... xvi

1 . INTRODUCTION .................................................................................................... 1

1.1 Background and Literature Review.......................................................................... 1 1.2 Objective and Scope ................................................................................................. 8

2 . WAVE LOADS ON FLOATING PLATFORM .................................................... 11

2.1 Introduction ............................................................................................................ 11 2.2 Wave Theory .......................................................................................................... 11 2.3 Wave Loads on Structures...................................................................................... 14

2.3.1 Diffraction and Radiation Theory ................................................................... 14 2.3.2 First Order Boundary Value Problem ............................................................. 15 2.3.3 First Order Potential Forces ............................................................................ 18 2.3.4 Wave Loads in Time Domain ......................................................................... 20 2.3.5 Morison’s Formula .......................................................................................... 22

3 . DYNAMICS OF MOORING LINES .................................................................... 23

3.1 Introduction ............................................................................................................ 23 3.2 Theory of Rod ........................................................................................................ 24

4 . DYNAMICS OF HORIZONTAL AXIS WIND TURBINES ............................... 29

4.1 Introduction ............................................................................................................ 29 4.2 Mechanical Components and Coordinate Systems ................................................ 30 4.3 Blade and Tower Flexibility ................................................................................... 36 4.4 Kinematics .............................................................................................................. 37

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4.5 Generalized Active Forces ..................................................................................... 43 4.6 Generalized Inertia Forces ..................................................................................... 45 4.7 Kane’s Equations.................................................................................................... 46 4.8 Coupling between FAST and CHARM3D............................................................. 48

5 . DYNAMICS OF A MULTIPLE UNIT FLOATING OFFSHORE WIND TURBINE .................................................................................................................... 52

5.1 Introduction ............................................................................................................ 52 5.2 Equations of Motion ............................................................................................... 53

5.2.1 Generalized Active Force ................................................................................ 53 5.2.2 Generalized Inertia Force ................................................................................ 55

5.3 Coefficient Matrix of Kane’s Dynamics ................................................................ 57 5.4 Force Vector of Kane’s Dynamics ......................................................................... 59 5.5 Equations of Motion in FAST ................................................................................ 61 5.6 Consideration of the Shade Effect .......................................................................... 65

6 . CASE STUDY I: SINGLE-TURBINE HYWIND SPAR .......................................... 71

6.1 Introduction ............................................................................................................ 71 6.2 Numerical Model for 5MW Hywind Spar ............................................................. 72 6.3 Hydrodynamic Coefficients in the Frequency Domain .......................................... 75 6.4 Coupled Dynamic Analysis in the Time Domain .................................................. 78 6.4 Influence of Control Strategies .............................................................................. 88

6.4.1 Two Control Strategies .................................................................................... 88 6.4.2 Modification of Control Strategies .................................................................. 90 6.4.3 Response in Random Sea Environment .......................................................... 95

6.5 Discussion ............................................................................................................ 106

7 . CASE STUDY II: THREE-TURBINE SEMI-SEBMERSIBLE .............................. 109

7.1 Introduction .......................................................................................................... 109 7.2 Configuration of the Platform, Mooring System and Turbines ........................... 110 7.3 Hydrodynamic Coefficients in the Frequency Domain ........................................ 113 7.4 Time Marching Simulation in Normal Operational Condition ............................ 116

7.4.1 Viscous Damping Modeling .......................................................................... 117 7.4.2 System Identification and Free Decay Test ................................................... 118 7.4.3 Responses in Random Wind and Wave Environment ................................... 120

7.5 One Turbine Failure Simulation ........................................................................... 131 7.5.1 Blade Pitch Control Failure of One Turbine ................................................. 131 7.5.2 Partial Blade Broken Failure of One Turbine ............................................... 141 7.5.2 Full Blade Broken Failure of One Turbine ................................................... 153

7.6 Discussion ............................................................................................................ 163

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8 . CASE STUDY III: FIVE-TURBINE SEMI-SEBMERSIBLE ............................ 165

8.1 Introduction .......................................................................................................... 165 8.2 Configuration of the Platform, Mooring System and Turbines ........................... 165 8.3 Hydrodynamic Coefficients in Frequency Domain ............................................. 168 8.4 Time Marching Simulation in Normal Operational Condition ............................ 171

8.4.1 Viscous Damping Modeling .......................................................................... 171 8.4.2 System Identification and Free Decay Test ................................................... 172 8.4.3 Responses in Random Wind and Wave Environments ................................. 174

8.5 One Turbine Failure Simulation ........................................................................... 181 8.6 Discussion ............................................................................................................ 193

9 . CONCLUSION AND FUTURE WORK ............................................................. 195

9.1 Coupled Dynamic Analysis of MUFOWT .......................................................... 195 9.2 Future Work ......................................................................................................... 196

REFERENCES .......................................................................................................... 198

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LIST OF FIGURES

Page

Figure 1.1 Global cumulative installed wind capacity ................................................... 2

Figure 1.2 Size evolution of wind turbines over time .................................................... 2

Figure 1.3 Annual and cumulative wind installations by 2030 in the U.S. .................... 3

Figure 1.4 Concept design of Semi-submersible type MUFOWT ................................. 6

Figure 1.5 Semi-submersible type MUFOWT ............................................................... 7

Figure 1.6 Next generation wind farm ............................................................................ 8

Figure 3.1 Coordinate system for slender rod .............................................................. 25

Figure 4.1 Turbine reference frames (a) and reference points (b) ................................ 38

Figure 4.2 Tower deflection geometry ......................................................................... 40

Figure 4.3 Coupled hull and mooring, riser analysis .................................................... 48

Figure 4.4 Data transfer between CHARM3D and FAST ............................................ 51

Figure 5.1 Schematic configuration of MUFOWT ....................................................... 53

Figure 5.2 Tower base position vectors ........................................................................ 56

Figure 5.3 Coefficient matrix of single turbine and platform ....................................... 57

Figure 5.4 Coefficient matrix of multiple turbines and platform ................................. 58

Figure 5.5 Forcing vector of single turbine and platform ............................................. 60

Figure 5.6 Forcing vector of multiple turbines and platform ....................................... 61

Figure 5.7 Series of coefficient matrices ...................................................................... 62

Figure 5.8 Series of forcing vectors .............................................................................. 64

Figure 5.9 Wake turbulence behind individual wind turbines ...................................... 66

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Figure 5.10 Turbine spacing recommendation ............................................................... 67

Figure 5.11 Power production for downwind turbine .................................................... 68

Figure 5.12 Global wind field (a) and separated wind field (b) ..................................... 69

Figure 6.1 Hywind spar mooring line arrangement ...................................................... 75

Figure 6.2 Discretized panel model of floating body (Hywind spar) ........................... 75

Figure 6.3 Normalized mode shapes of (a) tower fore-aft, (b) tower side-to-side and (c) blades .............................................................................................. 77

Figure 6.4 Surge motion (a) and spectra (b) (Hywind spar) ......................................... 80

Figure 6.5 Sway motion (a) and spectra (b) (Hywind spar) ......................................... 81

Figure 6.6 Heave motion (a) and spectra (b) (Hywind spar) ........................................ 81

Figure 6.7 Roll motion (a) and spectra (b) (Hywind spar) ........................................... 81

Figure 6.8 Pitch motion (a) and spectra (b) (Hywind spar) .......................................... 82

Figure 6.9 Yaw motion (a) and spectra (b) (Hywind spar) ........................................... 82

Figure 6.10 Top view of mooring-line arrangement (Hywind spar) .............................. 85

Figure 6.11 Top-tension (a) and spectra (b) of Line #1 (Hywind spar) ......................... 85



Figure 6.14 Tower-acceleration (a) and spectra (b) at 85.66m from MWL ................... 87



Figure 6.17 Two control strategies (Hywind spar) ......................................................... 89

Figure 6.18 Step wind input (a) and blade pitch angle (b) (Land-based) ....................... 91

Figure 6.19 Generator torque (a) and generator power (b) (Land-based) ...................... 92

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Figure 6.20 Step wind input (a) and blade pitch angle (b) (Hywind spar) ..................... 93

Figure 6.21 Platform pitch motion for two control strategies ........................................ 94

Figure 6.22 Generator torque (a) and generator power (b) for two control strategies .... 94

Figure 6.23 Blade pitch angle (a) and spectra (b) for two control strategies .................. 96

Figure 6.24 Shaft thrust force (a) and spectra (b) for two control strategies .................. 97

Figure 6.25 Surge motion (a) and spectra (b) for two control strategies ........................ 98

Figure 6.26 Sway motion (a) and spectra (b) for two control strategies ........................ 98

Figure 6.27 Heave motion (a) and spectra (b) for two control strategies ....................... 98

Figure 6.28 Roll motion (a) and spectra (b) for two control strategies .......................... 99

Figure 6.29 Pitch motion (a) and spectra (b) for two control strategies ......................... 99

Figure 6.30 Yaw motion (a) and spectra (b) for two control strategies .......................... 99

Figure 6.31 Rotor speed (a) and spectra (b) for two control strategies ........................ 101

Figure 6.32 Generator power (a) and spectra (b) for two control strategies ................. 101

Figure 6.33 Tower-base fore-aft shear force (a) and spectra (b) for two control strategies ........................................................................... 102

Figure 6.34 Tower-base axial force (a) and spectra (b) for two control strategies ....... 102

Figure 6.35 Tower-base pitch bending moment (a) and spectra (b) for two control strategies ........................................................................... 103

Figure 6.36 Flapwise shear force at blade root (a) and spectra (b) for two control strategies ........................................................................... 103

Figure 6.37 Edgewise shear force at blade root (a) and spectra (b) for two control strategies ........................................................................... 104

Figure 6.38 Top-tension of Line #1 (a) and spectra (b) for two control strategies ....... 105

Figure 6.39 Top-tension of Line #2 (a) and spectra (b) for two control strategies ....... 105

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Figure 6.40 Spectra of top-tension of Line #1 (a) and Line #2 (b) for quasi-static and FE mooring ................................................................ 106

Figure 7.1 Triangular platform geometry (a) and system configuration (b) ............... 110

Figure 7.2 Platform dimensions from top (a) and side (b) (Triangular platform) ...... 111

Figure 7.3 Mooring line configurations (Triangular platform) .................................. 113

Figure 7.4 Discretized panel model of floating body (Triangular platform) .............. 114

Figure 7.5 Added mass (Triangular platform) ............................................................ 114

Figure 7.6 Radiation damping (Triangular platform) ................................................. 115

Figure 7.7 Free decay test (Triangular platform) ........................................................ 119

Figure 7.8 Surge motion (a) and spectrum (b) (Triangular platform) ........................ 122

Figure 7.9 Sway motion (a) and spectrum (b) (Triangular platform) ......................... 122

Figure 7.10 Heave motion (a) and spectrum (b) (Triangular platform)........................ 123

Figure 7.11 Roll motion (a) and spectrum (b) (Triangular platform) ........................... 123

Figure 7.12 Pitch motion (a) and spectrum (b) (Triangular platform) ......................... 124

Figure 7.13 Yaw motion (a) and spectrum (b) (Triangular platform) .......................... 124

Figure 7.14 Generator power (a) and spectra (b) (Triangular platform) ...................... 125

Figure 7.15 Blade pitch angle (a) and spectra (b) (Triangular platform) ..................... 126

Figure 7.16 Tower base fore-aft shear force (a) and spectra (b)................................... 127

Figure 7.17 Tower base axial force (a) and spectra (b) (Triangular platform) ............. 128

Figure 7.18 Tower base pitch moment (a) and spectra (b) (Triangular platform) ........ 129

Figure 7.19 Tower base torsional moment (a) and spectra (b) (Triangular platform) .. 130

Figure 7.20 Turbine location (a) and turbine failure (b) (Triangular platform) ........... 131

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Figure 7.21 Blade pitch angle (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 132

Figure 7.22 Platform translational motions (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 133

Figure 7.23 Platform rotational motions (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 134

Figure 7.24 Generator power (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 135

Figure 7.25 Tower base fore-aft shear force (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 136

Figure 7.26 Tower base pitch moment (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 137

Figure 7.27 Tower base torsional moment (a) and spectra (b) with blade control failure (Triangular platform) ..................................................................... 138

Figure 7.28 Top view of mooring-line arrangement (Triangular platform) ................. 139

Figure 7.29 Top-tension (a) and spectra (b) of Line #1~#3 (Triangular platform) ...... 139

Figure 7.30 Top-tension (a) and spectra (b) of Line #4~#6 (Triangular platform) ...... 140

Figure 7.31 Blade length and broken zone ................................................................... 143

Figure 7.32 Tower base side-to-side shear force (a) and spectra (b) with partially broken blade ................................................................................ 144

Figure 7.33 Tower base roll moment (a) and spectra (b) with partially broken blade ................................................................................ 146

Figure 7.34 Tower base pitch moment (a) and spectra (b) with partially broken blade ................................................................................ 147

Figure 7.35 Tower base torsional moment (a) and spectra (b) with partially broken blade ................................................................................ 150

Figure 7.36 Top-tension (a) and spectra (b) of Line #1~#3 with partially broken blade ................................................................................ 151

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Figure 7.37 Top-tension (a) and spectra (b) of Line #4~#6 with partially broken blade ................................................................................ 152

Figure 7.38 Tower base fore-aft shear force (a) and spectra (b) with fully broken blade ...................................................................................... 154

Figure 7.39 Tower base side-to-side shear force (a) and spectra (b) with fully broken blade ...................................................................................... 156

Figure 7.40 Tower base pitch moment (a) and spectra (b) with fully broken blade ..... 158

Figure 7.41 Tower base torsional moment (a) and spectra (b) with fully broken blade ...................................................................................... 159

Figure 7.42 Transition of the side-to-side excitation of turbine #1 .............................. 160

Figure 7.43 Top-tension (a) and spectra (b) of Line #1~#3 with fully broken blade ... 161

Figure 7.44 Top-tension (a) and spectra (b) of Line #4~#6 with fully broken blade ... 162

Figure 8.1 Rectangular platform geometry (a) and system configuration (b) ............ 166

Figure 8.2 Platform dimensions from top (a) and side (b) (Rectangular platform) .... 166

Figure 8.3 Mooring line configuration (Rectangular platform) .................................. 168

Figure 8.4 Discretized panel model of floating body (Rectangular platform) ........... 168

Figure 8.5 Added mass (Rectangular platform) ......................................................... 169

Figure 8.6 Radiation damping (Rectangular platform) ............................................... 170

Figure 8.7 Free decay test (Rectangular platform) ..................................................... 172

Figure 8.8 Surge motion (a) and spectrum (b) (Rectangular platform) ...................... 174

Figure 8.9 Sway motion (a) and spectrum (b) (Rectangular platform) ...................... 175

Figure 8.10 Heave motion (a) and spectrum (b) (Rectangular platform) ..................... 175

Figure 8.11 Roll motion (a) and spectrum (b) (Rectangular platform) ........................ 175

Figure 8.12 Pitch motion (a) and spectrum (b) (Rectangular platform) ....................... 176

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Figure 8.13 Yaw motion (a) and spectrum (b) (Rectangular platform) ........................ 176

Figure 8.14 Generator power (a) and spectra (b) (Rectangular platform) .................... 177

Figure 8.15 Blade pitch angle (a) and spectra (b) (Rectangular platform) ................... 178

Figure 8.16 Tower base fore-aft shear force (a) and spectra (b)................................... 179

Figure 8.17 Tower base pitch moment (a) and spectra (b) (Rectangular platform) ..... 180

Figure 8.18 Tower base torsional moment (a) and spectra (b) ..................................... 181

Figure 8.19 Turbine location (Rectangular platform) ................................................... 182

Figure 8.20 Blade pitch angle (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 183

Figure 8.21 Platform translational motions (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 184

Figure 8.22 Platform rotational motions (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 185

Figure 8.23 Generator power (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 186

Figure 8.24 Tower base fore-aft shear force (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 188

Figure 8.25 Tower base pitch moment (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 189

Figure 8.26 Tower base torsional moment (a) and spectra (b) with blade control failure (Rectangular platform) ................................................................... 190

Figure 8.27 Top view of mooring-line arrangement (Rectangular platform) ............... 191

Figure 8.28 Top-tension (a) and spectra (b) with blade control failure ........................ 192

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LIST OF TABLES

Page Table 4.1 Coordinate system of wind turbine ............................................................. 31

Table 4.2 Degree of freedom variables ....................................................................... 34

Table 6.1 Specification of 5MW turbine ..................................................................... 73

Table 6.2 Specification of Hywind spar platform ....................................................... 73

Table 6.3 Specification of Hywind spar mooring system ........................................... 74

Table 6.4 Natural frequencies of platform motions (Hywind spar) ............................ 77

Table 6.5 Natural frequencies of tower and blade at 17.11 m/s steady wind .............. 78

Table 6.6 Wind load for uncoupled dynamics ............................................................ 79

Table 6.7 Environmental conditions (Hywind spar) ................................................... 79

Table 6.8 Platform motion statistics (Hywind spar) .................................................... 83

Table 6.9 Mooring line top tension statistics (Hywind spar) ...................................... 86

Table 6.10 Tower acceleration statistics (Hywind spar) ............................................... 88

Table 6.11 Statistics of platform motion in two control strategies ............................. 100

Table 7.1 Specification of triangular platform .......................................................... 111

Table 7.2 Specification of mooring system (Triangular platform)............................ 112

Table 7.3 Drag coefficients of Morison members (Triangular platform) ................. 118

Table 7.4 Natural frequencies of triangular platform ................................................ 120

Table 7.5 Environmental conditions (Triangular platform) ...................................... 121

Table 7.6 Blade structural properties ........................................................................ 141

Table 7.7 Damaged blade length and node number .................................................. 142

xvii

Table 7.8 Tower base side-to-side shear force statistics with partially broken blade .......................................................................................................... 145

Table 7.9 Tower base roll moment statistics with partially broken blade ................. 147

Table 7.10 Tower base pitch moment statistics with partially broken blade .............. 148

Table 7.11 Tower base torsional moment statistics with partially broken blade ........ 149

Table 7.12 Mooring line top tension statistics with partially broken blade ................ 153

Table 7.13 Tower base fore-aft shear force statistics with fully broken blade............ 155

Table 7.14 Tower base side-to-side shear force statistics with fully broken blade ..... 156

Table 7.15 Tower base pitch moment statistics with fully broken blade .................... 157

Table 7.16 Tower base torsional moment statistics with fully broken blade .............. 160

Table 7.17 Mooring line top tension statistics with fully broken blade ...................... 163

Table 8.1 Specification of rectangular platform ........................................................ 167

Table 8.2 Drag coefficients of Morison members (Rectangular platform) ............... 172

Table 8.3 Natural frequencies of rectangular platform ............................................. 173

1

1. INTRODUCTION

1.1 Background and Literature Review

During the past century, people have depended on fossil fuels as their major

source of energy. However, fossil fuels continue to be depleted and their negative

environmental impact is very alarming. Therefore, the importance of increasing the use

of clean renewable energy cannot be over emphasized for the secure future of all human

beings. As a result, the number of wind turbines has rapidly increased all around the

world. Wind energy resources have many benefits. The first and most important benefit

is that wind energy is economically competitive. Today’s rising oil and gas prices are a

serious threat to the economies and industries in the United States, so building a new

wind plant is the most competitive way to produce a new electricity generation source.

Unlike most other energy resources, wind turbines do not consume water while fossil

fuel and nuclear energy plants require large amounts of water for their cooling systems.

Wind energy is inexhaustible and infinitely renewable and does not produce any carbon

emissions.

As can be seen in Figure 1.1, the global cumulative wind energy capacity

gradually increased over the year, and reached 237.7GW in 2011. The turbine size is

also growing as the capacity increases in Figure 1.2. In the 1990’s, the largest turbine

produced 2MW and the diameter of the rotor was approximately 80m. Recently, the

capacity has increased up to 8 ~ 10MW and the size of rotors has doubled to around

160m.

2

Figure 1.1 Global cumulative installed wind capacity (Global Wind Energy Council, 2011)

Figure 1.2 Size evolution of wind turbines over time (EWEA, 2010)

In the case of the United States, 5,116MW of wind power was created in 2010

and over 5,600MW of wind power is currently under construction. Total U.S. wind

installations stand at 40,181MW, which represents 21% of global wind capacity.

Furthermore, the U.S. government expects that wind energy will produce 20% of total

energy by 2030. To implement the 20% wind scenario, new wind power installations

3

would increase to more than 16,000MW per year by 2018, and continue at that rate

through 2030 as shown in Figure 1.3.

Figure 1.3 Annual and cumulative wind installations by 2030 in the U.S. (Courtesy of http://www.20percentwind.org/)

However, the on-land wind farms also have many negative features such as lack

of available space, noise restriction, shade, visual pollution, limited accessibility in

mountainous areas, community opposition, and regulatory problems. Therefore, many

countries in Europe have started to build wind turbines in coastal waters, and so far most

offshore wind farms have been installed in relatively shallow-water areas less than 40m

deep by using bottom-fixed-type base structures.

Recently, several countries have started to plan offshore floating wind farms.

Although they are considered to be more difficult to design, wind farms in deeper waters

are, in general, less sensitive to space availability, noise restriction, visual pollution, and

regulatory problems. They are also exposed to much stronger and steadier wind fields to

4

become more effective. Furthermore, in designing these floating wind farms, the existing

technology and experience of offshore industry used for petroleum production has been

helpful. In this regard, if technology and infrastructure is fully developed, offshore

floating wind farms are expected to produce huge amounts of clean electricity at

competitive prices compared to other energy sources (Henderson et al., 2002; Henderson

et al., 2004; Musial et al., 2004; Tong, 1998; Wayman et al., 2006). Possible

disadvantages of floating type wind farms include the complexity of blade controls due

to platform motion, and a larger inertia loading on the tall tower caused by greater

floater accelerations, etc. They are also directly exposed to the open ocean without any

natural protection so they may have to endure harsher environments.

On the other hand, there are also merits of floating bases compared to fixed bases

in the dynamic/structural point of view. In case of fixed offshore wind turbines (OWTs),

the high-frequency excitations caused by rotating blades and tower flexibility may cause

resonance at the system’s natural frequencies. This is particularly so as water depth

increases which may significantly shorten its fatigue life. For floating wind turbines,

however, their natural frequencies of 6-DOFs motion are typically much lower than

those rotor-induced or tower-flexibility-induced excitations (Roddier et al., 2009), so the

possibility of dynamic resonance with the tower and blades is much less (Jonkman and

Sclavounos, 2006; Withee, 2004). The TLP-type OWT is one exception (Bae et al., 2010;

Jagdale and Ma, 2010). TLP-types are much stiffer in the vertical-plane modes

compared to other floating wind turbines, so the effects of such high-frequency

excitations from the tower and blades need to be checked.

5

Another concept for floating offshore wind farms is the Multiple Unit Floating

Offshore Wind Turbine (MUFOWT). This model includes multiple turbines upon a

single floating platform rather than the typical concept of the floating offshore wind

turbine (FOWT) where each turbine has its own floating platform. The possible

advantages and disadvantages of MUFOWT over single floating turbine were checked

(Barltrop, 1993), and an effort was made to develop analytical tools for evaluating the

performance of multiple turbine wind farm was made (Henderson et al., 1999)

Compared to the single unit floating wind turbine, MUFOWT has several

advantages. It may reduce installation cost because only one mooring system is

necessary for multiple turbines. From a stability point of view, MUFOWT provides a

more stable condition than a single unit structure. This characteristic also enables higher

towers and better energy capture. Better platform response in random sea environments

can also be ensured because larger floating units usually tend to have less response. The

easy access to MUFOWT is also one of the advantages compared to the single turbine

unit. A larger floating platform may be equipped with a helicopter landing deck, so

access by air can be available. On the other hand, there are also several disadvantages of

the MUFOWT concept. One of the most serious problems is the interference between

turbines, and possibility of performance drop due to the shade effect of the blades or

tower. This disadvantage can be overcome by adopting a specific design of floater or

arrangement of turbines. In some point, weathervaning design of floater is necessary to

avoid excessive interference between turbines. In addition to the interference problem,

6

the MUFOWT is not suitable for shallow water area, because mooring design and power

transport in shallow water depth will be difficult.

At the earlier stage of research about MUFOWT, such a large floating structure

and multiple turbines are not regarded as cost-effective. However, technological

developments and recent trends in the rapid increase in size of wind turbines make this

concept more viable.

Figure 1.4 Concept design of Semi-submersible type MUFOWT

(Henderson and Patel, 2003) Figure 1.4 shows an example of a semi-submersible type MUFOWT which is

suggested by a research project conducted by University College London (Henderson

and Patel, 2003). This MUFOWT concept has five turbines upon on large semi-

submerged pontoon. In order to minimize wave loads and the resulting platform motion,

the main structure is located below the sea surface. It also has one turret mooring system

at the center of the structure, and each turbine is located symmetrically about the anchor

7

position. This mooring concept enables the whole structure to rotate so that the series of

turbines face the wind direction.

Similar semi-submersible types, but with different hull forms are also suggested

(Henderson et al., 2000). Both weathervaning and non-weathervaning vessels are

suggested as shown in Figure 1.5. As pointed out earlier, the weathervaning vessel

always faces into the wind so a relatively expensive turret mooring system is required.

On the other hand, a non-weathervaning model cannot rotate to face into the wind, and

the possibility of wake interference from another turbine is very high. However, this

model is more cost-effective compared to the weathervaning model. In order to

compromise minimal fatigue loads and minimum pontoon cost, one of the fractal designs

was recommended.

(a) Weathervaning model (b) Non-weathervaning model

Figure 1.5 Semi-submersible type MUFOWT (Henderson et al., 2000) Recently, research concerning the feasibility for three turbines on one floating

unit was carried out, and the model tests were performed (Lefranc and Torud, 2011). As

can be seen in Figure 1.6, the proposed model has three inclined towers and the mooring

8

lines are connected to a turret at the vessel’s geometric center. This research showed that

the concept design proved to be feasible in water depth from 45m and deeper. Regarding

cost effectiveness, the concept is comparable to today’s solution when it comes to the

cost of energy production.

Figure 1.6 Next generation wind farm (Courtesy of http://www.windsea.no/) 1.2 Objective and Scope

The main objective of this research is to develop a coupled dynamic analysis tool

for MUFOWT. As mentioned earlier, an analysis tool for large floating offshore wind

farms was developed (Henderson, 2000) in the state/frequency domain and several

different designs of MUFOWT were suggested. The developed tools were primarily

used to obtain motion responses and loads for several locations of platform structures.

However, these analysis tools did not include the mooring line analysis tool, hence the

dynamic coupling effect between hull and mooring lines could not be accounted for.

9

Moreover, the turbine model used in those analysis tools was able to estimate fatigue

damage but did not consider elasticity of tower and blades; this is proved to be very

important to the response of turbine and platform. So, the effectiveness of analysis tools

in this research was very limited.

Another design code which has been developed for years in the U.S. wind energy

industry is FAST. The National Renewable Energy Laboratory (NREL) and its academic

and industry partners have created aero-elastic simulators for horizontal axis wind

turbines for both two- and three-bladed turbines. This design code was initially

developed with a built-in aerodynamics code (Wilson et al., 1995) and later merged with

the AeroDyn subroutine library of rotor-aerodynamics routines developed by NREL to

compute the aerodynamic forces on the turbine blades (Wilson et al., 2000). Recently,

FAST has been updated to accommodate a greater degree of freedom of turbines and

hydrodynamic calculations for floating offshore wind turbines (Jonkman and Buhl Jr,

2005). So far, many of the wind energy industries in the world have used this FAST

design code to evaluate their turbines and the code has been verified by the world’s

foremost certifying body for wind turbines, Germanischer Lloyd (GL) WindEnergie

GmbH, located in Hamburg, Germany (Manjock, 2005). Since FAST can model many

common turbine configurations and flexible elements using modal representation and

analyze in the time domain, it is considered to be the most advanced design code in the

wind energy industry to date.

However, the most up-to-date version of FAST has been optimized for land-

based turbines or single-turbine floating wind farm analysis, so direct use of FAST for

10

analyzing MUFOWT is impossible. Furthermore, the hydrodynamic module inside

FAST has several limitations and the mooring module can deal only with the quasi-static

model of lines, so the dynamic behavior of mooring lines cannot be calculated during

time domain simulation.

For this reason, a portion of the FAST algorithm is implemented into the floater-

mooring coupled dynamic analysis program, CHARM3D, and vice versa so that the

tower-floater coupling can be accurately achieved. By combining FAST and CHARM3D,

the wind turbine design code FAST can include a more efficient hydrodynamic module

and finite element dynamic mooring line module at the same time.

To achieve the objectives of this study, the combined design code FAST-

CHARM3D will be extended in order to accommodate multiple turbines on a single

floating platform. Of course, multiple turbine dynamics should be solved simultaneously

with the time marching scheme as done by single turbine analysis. The research and

development works which will be covered in this thesis will include

Dynamic coupling between FAST and CHARM3D

Development of coupled dynamic analysis tools for MUFOWT

Design verification of MUFOWT platforms in time domain

Dynamic load analysis in turbine failure conditions.

11

2. WAVE LOADS ON FLOATING PLATFORM

2.1 Introduction

In this section the wave loads and dynamic responses of floating platform are

reviewed. In the beginning, the wave theory of first order is reviewed, and the review of

diffraction theory with first order potential force and moment on floating platform

follows. Finally, the Morison’s formula for inertia and drag force in the time domain will

be also presented.

2.2 Wave Theory

To derive wave theory, Boundary Value Problem (BVP) with proper kinematic

and dynamic boundary conditions needs to be solved. The governing equation of fluid

with assumption of irrotational, incompressible and inviscid properties can be defined by

Laplace’s equation:

2 0 (2.1)

To solve the Laplace equation, the proper boundary conditions in the domain

should be defined. The common boundary conditions for ocean water wave problems are

introduced and explained. On the free surface, water waves should satisfy two boundary

conditions; kinematic and dynamic boundary conditions. The kinematic boundary

condition indicates that water particles on the free surface should remain on the free

surface and be formulated as below.

12

0u vt x y t

at , ,z x y t (2.2)

where , ,x y t is the free surface elevation written in the spatial and time domain. The

dynamic free surface boundary condition states that the pressure on the free surface must

be the same as the atmospheric pressure and be constant along the free surface.

2 2 210

2 x y z gzt

at , ,z x y t (2.3)

For bottom boundary condition, the vertical component of velocity of fluid particle at the

ocean bottom is zero and formulated as below.

0z

at z d (2.4)

where d is water depth. This boundary condition represents that the water particles at

the bottom cannot penetrate the ocean bottom.

The exact solution of the Laplace equation with the given boundary conditions

above is difficult to obtain due to nonlinear terms of the free surface boundary

conditions. So the perturbation method with small wave amplitude assumption can be

used to obtain an approximated solution of a certain order of accuracy. The following

equations show the first and second order velocity potentials and free surface elevations.

First order velocity potential and free surface elevation:

(1) ( cos sin )coshRe

coshi kx ky tk z digA

ekd

(2.5)

(1) cos cos sinA kx ky t (2.6)

Second order velocity potential and free surface elevation:

13

(2) 2 (2 cos 2 sin 2 )4

cosh 23Re

8 sinhi kx ky tk z d

A ekd

(2.7)

(2) 23

coshcos 2 cos 2 sin 2

sinh

kdA kx ky t

kd (2.8)

where A is the wave amplitude, is the wave frequency, k is the wave number, and

is the incident wave heading angle.

In a random sea environment, a fully developed wave condition can be modeled

as wave spectra by summation of regular wave trains and random phases. In the ocean

engineering field, various wave spectra such as JONSWAP (Joint North Sea Wave

Observation Project) and Pierson-Moskowitz are proposed and used. The simulated

random wave time series from the given wave spectrum ( )S can be expressed by

superposition of a large number of linear wave components with random phases.

( )

1 1

, cos Re i i i

N Ni k x t

i i i i ii i

x t A k x t Ae

(2.9)

2 ( )i iA S (2.10)

where, N and are the number of wave components and intervals of frequency

division, and i is a random phase angle generated by random function. To avoid the

repetition of random wave realization with a limited number of wave components, some

modification was made and re-written as below.

'( )

1

( , ) Re i ii

Ni k x t

ii

x t Ae

(2.11)

14

where 'i i i and i is the random perturbation number uniformly distributed

between 2 and 2 .

2.3 Wave Loads on Structures

It is important to predict wave loads on a structure in studying the dynamics of

the floating platform. In deeper water, the diffraction of waves around the platform is

significant. So, the diffraction theory is the most appropriate way to describe the wave

loading on the platform. In case of a slender body, Morison’s formula is also widely

used. In extreme environmental conditions, viscous force may become important and

should be taken into consideration. In this section, both the diffraction theory and

Morison’s formula are discussed and these will be used to compute the wave load in our

study.

2.3.1 Diffraction and Radiation Theory

To see the interaction between incident waves and large floating structures, the

boundary value problem is reviewed in this section. As we already mentioned, the total

velocity potential satisfies Laplace equations, free surface boundary conditions, and

the bottom boundary condition. This total velocity potential includes the incident

potential I , diffraction potential D and radiation potential R and can be expressed

by a perturbation series with respect to the wave slope parameter

( ) ( ) ( ) ( )

1 1

n n n n n nI D R

n n

(2.12)

15

where, ( )n represents the n th order solution of , and solutions up to second order

will be considered in this study.

To solve the wave and floating body interaction problem, an additional boundary

condition, which is called body boundary condition, should also be considered. By

introducing surface normal vector n , the body boundary condition can be expressed as

nV

n

on body surface (2.13)

where, nV is the normal velocity vector of the body at its surface.

In addition, the diffraction ( D ) and radiation potential ( R ) also should satisfy

the Sommerfeld radiation condition at the far field boundary.

,,lim 0D R

D Rr

r ikr

(2.14)

where, r is the radial distance from the center of the floating body.

2.3.2 First Order Boundary Value Problem

The first order interaction of a monochromatic incident wave with a freely

floating body will be reviewed in this section. The first order potential can be re-written

by separating the time dependency explicitly as

(1) (1) (1) (1) (1) (1) (1)I D R = Re ( , , ) ( , , ) ( , , ) i t

I D R x y z x y z x y z e (2.15)

The first order incident potential (1)I is the linear wave potential re-written as

(1) cosh( ( ))( , , ) Re

cosh( )i

I

igA k z dx y z e

kd

k x (2.16)

16

where, K is a vector wave number with Cartesian components ( cos , sin , 0)k k , and

x is the position vector in the fluid. Here is the angle of the incident wave relative to

the positive x axis.

So, the boundary value problem governing the first order diffraction and

radiation potentials can be summarized as

2 (1), 0D R in the fluid ( 0z ) (2.17)

2 (1), 0D Rg

z

on the free surface ( 0z ) (2.18)

(1), 0D R

z

on the bottom ( z d ) (2.19)

(1)

(1) (1)R in

n ξ α r on the body surface (2.20)

(1),lim 0D R

rr ik

r

at far field (2.21)

where r represent the position vector on the body surface, r is the radial distance from

the origin and n is the unit normal vector pointing into the fluid domain at the body

surface. The first-order motion of the body in the translational ( (1)Ξ ) and rotational ( (1) )

directions have the forms

(1) (1)Re i te Ξ ξ (1) (1) (1) (1)1 2 3, , ξ (2.22)

(1) (1)Re i te α (1) (1) (1) (1)1 2 3, , α (2.23)

17

where the subscripts 1,2 and 3 denote the translational (surge, sway and heave) and

rotational (roll, pitch and yaw) modes with respect to the x , y and z axes respectively.

The six degrees of freedom of first order motion can be also simplified as

(1)i i for 1, 2,3i (2.24)

(1)3i i for 4,5,6i (2.25)

Based on that motion, the radiation potential, which represents the fluid

disturbance due to the motion of the body, can be further decomposed as

6(1) (1)

1R i i

i

(2.26)

where i represent the first order velocity potential of the rigid body motion with unit

amplitude in the i th mode without incident waves. The body boundary condition of each

mode can be also expressed by replacing (1)i

(1)i

inn

1, 2,3i (2.27)

(1)

3i

in

r n 4,5,6i (2.28)

on the body surface.

The first order diffraction potential (1)D represents the disturbance to the incident

wave due to the presence of the body in its fixed position. This velocity potential should

satisfy the body surface boundary condition below.

(1) (1)D I

n n

on the body surface (2.29)

18

2.3.3 First Order Potential Forces

Now, the first order hydrodynamic potential force on the floating structure can be

obtained by solving the first order diffraction ( (1)D ) and radiation ( (1)

R ) potentials. By

adopting the perturbation method, the hydrodynamic pressure P t can be expressed as

(1)(1)P

t

(2.30)

The total force and moment on the body can be directly obtained by simple

integration over the instantaneous wetted body surface S t .

= 1,2,3

= 4,5,6

b

b

i

S

i

iS

Pn dS i

tP dS i

F

r n (2.31)

where, bS is the body surface at rest.

The first order total force and moment including the hydrostatic term can now be

expressed as

(1) (1) (1) (1)HS R EX F F F F (2.32)

where subscript HS represents the hydrostatic restoring force and moment, R represents

the force and moment from the radiation potential, and EX represents the wave exciting

force and moment from the incident and diffraction potentials.

The hydrostatic restoring force and moment ( (1)HSF ) are induced by hydrostatic

pressure change due to the motion of the body. It can be expressed as

(1) (1){ }HS F K (2.33)

19

where, (1) is the first order motion of the floating body, and K represents the

hydrostatic restoring stiffness.

The force and moment from the radiation potential ( (1)RF ) comes from the first

order motions of the floating body. The radiation potential ( (1)R ) induces the added mass

and radiation damping, and it can be expressed as

(1) (1)ReR F f (2.34)

where

B

iij j

S

f dSn

f , 1, 2, ,6i j (2.35)

The set of coefficients ijf is complex and the real and imaginary parts are

dependent on the frequency . The coefficients can be re-written as

2 aij ij ijf M i C (2.36)

So, the force and moment from the radiation potential can be expressed as

(1) (1) (1)Re aR F M C (2.37)

where, aM is the added mass coefficients and C is the radiation damping coefficients.

The last term (1)EXF represents the first order wave excitation forces and moments,

which are derived by incident and diffraction wave potentials. It can be written as

0

(1) Re ji tEX I D

S

Ae dSn

F 1, 2, , 6j (2.38)

20

where, A is wave amplitude. It is seen that the first order wave excitation forces and

moments are proportional to wave amplitude which is frequency dependent. The first

order wave exciting force from the unit wave amplitude is called Linear Transfer

Function (LTF) which represents the relationship between incident wave elevation and

the first order diffraction forces on the floating body.

2.3.4 Wave Loads in Time Domain

In this section, the time domain realization of the first and second wave forces

and moments in a random sea environment will be presented. The first and second order

hydrodynamic forces and moments on a body due to stationary Gaussian random seas

can be expressed as a two-term Volterra series in the time domain as below

(1) (2)1 2 1 2 1 2 1 2,t t h t d h t t d d

F F (2.39)

where t is the ambient wave free surface elevation at a reference position, 1h is

the linear impulse response function, and 2 1 2, h is the quadratic impulse response

function. The above equation can be rewritten in the form of the summation of the N

frequency components as follows

(1)

1

ReN

i tI j j

j

t A e

F L (2.40)

(2) *

1 1 1 1

Re , ,N N N N

i t i tI j k j k j k j k

j k j k

t A A e A A e

F D S (2.41)

21

where jL represents the linear force transfer function (LTF), and ,j k D and

,j k S are the difference and the sum frequency quadratic transfer functions (QTF),

respectively.

The time domain expression from radiation potential forces and moments has the

following form

t

aR t t t τ d

F M R (2.42)

where a M is the added mass at infinite frequency, and tR is called a retardation

function or time memory function which is related to the frequency domain solution of

the radiation problem as follows

0

2 sin tt C d

R (2.43)

where C is the radiation damping coefficient in the Equation (2.36) at frequency .

The total wave loads in the time domain can now be obtained by summing each

force component as follows

Total I Rt t t F F F (2.44)

where (1) (2)Total t t t F F F is the total wave exciting force,

(1) (2)I I It t t F F F is the sum of the Equation (2.40) and (2.41), R tF is the

radiation term from the Equation (2.42).

22

2.3.5 Morison’s Formula

For slender cylindrical structures, the diffraction effect is usually negligible and

the viscous effect becomes dominant. In that case, the inertia effect including the added

mass and damping effect can be simply estimated by Morison’s formula (Morison et al.,

1950). This formula states that the wave load per unit length of the structure normal to

the elemental section with diameter D which is small compared to the wave length is

obtained by the sum of an inertial, added mass and drag force.

2 2 1

4 4 2m m n a n D S n n n n

D DF C u C C D u u

(2.45)

where mF denotes Morison’s force, am CC 1 is the inertia coefficient, aC is the

added mass coefficient, DC is the drag coefficient, SD is the breadth or diameter of the

structure, nu and nu are the acceleration and the velocity of the fluid normal to the

body, respectively, and n and n are the normal acceleration and the velocity of the

body, respectively. The first two terms in Equation (2.45) are the inertia forces from the

Froude-Krylov force and added mass effect. The last term represents the drag force in

the relative velocity form. This relative-velocity form indicates that the drag force

contributes to both the exciting force and damping force on the motion of the structure.

In this study, the viscous effects of slender members such as the cylindrical hull,

TLP columns or truss members are computed by Morison’s formula and are combined

with the potential forces to compute the wave forces on the platform.

23

3. DYNAMICS OF MOORING LINES

3.1 Introduction

In this chapter, the dynamics of the mooring lines and theoretical background

will be presented. The position of the floating wind turbine is maintained on the station

by mooring systems. Traditionally, ships were moored by anchor chains from the bow,

and floating production vessels such as FPSOs were moored by spread mooring which is

utilized by a turret mooring system or a Single Point Mooring (SPM) system. In the case

of a floating offshore platform such as TLP, taut vertical moorings or tendons, which are

usually made of steel pipes, have been used for the mooring system. Steel wire ropes

combined with a chain at each end also have been used for the spar platform. For ultra-

deep water around 3,000m depth, synthetic mooring lines such as polyester lines are

considered to be more efficient.

The basic concept of mooring systems for FOWTs is identical to the floating

offshore oil & gas platforms. Slack catenary, taut catenary, or taut tension-leg mooring

systems are considered to be common mooring systems in FOWTs. In addition to the

station-keeping purpose, the mooring systems also provide the platforms with stability.

For a TLP type platform design, the vertical tendons are main stability members so the

failure of the vertical tendon system would cause the failure of the complete system.

Therefore, the mooring system design of FOWT is one of the most important

components for the dynamic behavior of the entire system as well as its stability.

24

So far, not many studies have been done concerning the dynamic behavior of

mooring systems in FOWTs, and the effect of mooring systems has been estimated by

quasi-static catenary equations. In shallow waters, this quasi-static method shows

acceptable results because the total mass of mooring lines is negligible and the motion is

small. However for deep water, the dynamics of mooring lines, including line inertia and

the drag forces in the fluid, become more important, so finite-element mooring line

analysis, which can include those effects, has been adopted in this study and is definitely

preferred.

In this study, a three-dimensional elastic rod theory, which includes the line

stretching, has been adopted to model the mooring lines (Garrett, 1982). The finite

element method has been used to represent the analytic expression as a numerical form.

The rod theory has some advantages in that the governing equation has been developed

in a single global coordinate system and the geometric nonlinearity can be handled with

ease and efficiency.

3.2 Theory of Rod

This theory describes the behavior of slender rods in terms of the position of the

centerline of the rod. As illustrated in Figure 3.1, a position vector ( , )s tr is introduced

to define the space curve, which is a function of arc length s and time t . If we assume

that the rod is inextensible, then the unit tangent vector to the space curve is r , and the

principal normal vector is directed along r and the bi-normal is directed along r r

where the prime symbol represents the differentiation with respect to the arc length.

25

Figure 3.1 Coordinate system for slender rod

The equation of motion can be derived by equilibrium of the linear force and

moment at the unit arc length of the rod as follows

F q r (3.1)

M r F m 0 (3.2)

where q is the applied force per unit length, is the mass per unit length of the rod, m

is the applied moment per unit length. F and M are the resultant force and moment

along the centerline. The dot denotes the differentiation with respect to time.

The resultant moment M can be expressed as

EI H M r r r (3.3)

where EI is the bending stiffness, H is the torque. This relationship indicates that the

bending moment is proportional to the curvature and is directed along the bi-normal

direction. Substituting this relation into Equation (3.2), we have

26

EI H H r r F r r m 0 (3.4)

and the scalar product of the above equation with r yields

H m r 0 (3.5)

If we assume that there is no distributed torsional moment ( m r ), and the torque

in the lines is negligible, then the Equation (3.4) can be re-written as

EI r r F 0 (3.6)

Introducing a scalar function ( , )s t , which is called the Lagrangian multiplier,

the F in the above equation can be written as

EI F r r (3.7)

The scalar product of Equation (3.7) with r results in

EI F r r r (3.8)

or

2T EI (3.9)

where T is the tension and the is the curvature of the rod.

Combining Equation (3.7) with (3.1), the equation of motion for the rod become

EI r r q r (3.10)

In addition, r should satisfy the inextensible condition as

1 r r (3.11)

27

If the rod is extensible, and the stretch is linear and small, the above condition

(3.11) can be approximated by

11

2

T

AE AE

r r (3.12)

The above equation of motion of the rod and inextensible (or extensible)

condition with initial and boundary conditions and applied force vector q , are sufficient

to determine the position vector ( , )s tr and the Lagrangian multiplier ( , )s t . The

applied force vector q , in most offshore applications, comes from the gravity of the rod

and the hydrostatic and hydrodynamic forces from surrounding fluid. So it can be

expressed as

s dq w F F (3.13)

where w is the weight of the rod per unit length, sF is the hydrostatic force and dF is the

hydrodynamic force on the rod per unit length. The hydrostatic force can be written as

P sF B r (3.14)

where B is the buoyancy force on the rod per unit length, and P is the hydrostatic

pressure at point r on the rod.

The hydrodynamic force dF can be derived by Morison’s formula below

n n n n n nA M D

nA

C C C

C

d

d

F r V V r V r

r F

(3.15)

where AC is the added mass coefficient per unit length, MC is the inertial coefficient per

unit length per unit normal acceleration and DC is the drag coefficient per unit length per

28

unit normal velocity. nV and nV are fluid velocity and acceleration normal to the rod

centerline respectively. They can be expressed as

n V V r V r r r (3.16)

n V V V r r (3.17)

where V and V are the total fluid particle’s acceleration and velocity at the centerline of

the rod without disturbance by the rod. nr and nr are the rod acceleration and velocity

normal to its centerline and can be obtained by

n r r r r r (3.18)

n r r r r r (3.19)

The equation of motion of the rod subjected to its weight, hydrostatic and

hydrodynamic forces in water, combining Equations (3.13) through (3.15) with (3.1)

becomes

dna wC EI r r r r w F (3.20)

where

2 2T P EI T EI (3.21)

w w B (3.22)

T T P (3.23)

T is the effective tension in the rod, w is the effective weight or the wet weight.

The Equation (3.20) together with the line stretch condition in Equation (3.12),

are the governing equations for the statics or dynamics of the rod in fluid.

29

4. DYNAMICS OF HORIZONTAL AXIS WIND TURBINES

4.1 Introduction

The dynamics response of three-bladed, horizontal axis wind turbines (HAWT)

can be analyzed by structural modeling with proper geometry, coordinate systems and

degrees of freedom (DOFs). Thus, accurate structural models are necessary to analyze

wind energy systems. To deal with multiple components of wind turbines such as

floating platform, towers, blades and Rotor Nacelle Assembly (RNA), Kane’s method

(originally called Lagrangian form of d’Alembert’s principle) is used to set up equations

of motion which can be handled by numerical integration. This method can greatly

simplify the equations of motion. Consequently the equations are easier to solve than

other dynamic approaches using methods of Newton or Lagrange. Furthermore,

computation time can be also reduced by using fewer terms than other conventional

approaches. This chapter revisits the steps to establish the equations of motion for

HAWT which is employed by computational design code FAST (Jonkman, 2003;

Wilson et al., 2000). First, wind turbine geometry with various rigid bodies is defined,

and then coordinate systems and degrees of freedom are discussed. Since FAST models

the blades and tower as flexible bodies, the deflections and vibrations are presented with

the numerical model of elastic bodies. Aerodynamic load calculations, including

aerodynamic lift, drag, and pitching moment of the airfoil section along the wind turbine

blades, are carried out by AeroDyn, and the details of aerodynamics are not presented in

this study. Finally, the equations of motion, which describe the kinematic and kinetics of

30

wind turbine motion and force-acceleration relations of the entire wind turbine system,

are presented by Kane’s equations of motion.

4.2 Mechanical Components and Coordinate Systems

The FAST design code models a floating wind turbine with six rigid and five

flexible bodies. The six rigid bodies include the floating platform, nacelle, tower-top

base plate, armature, hub and gears. In detail, the tower is rigidly attached to the floating

platform and the top of the tower is fixed to a base plate which supports a yaw bearing

and nacelle. The nacelle assembly can be allowed to tilt and the low speed shaft (LSS)

connects the gearbox to the rotor. The rotor assembly consists of hub, blades, and tip

brakes. In terms of DOFs, platform rigid body motion accounts for six DOFs, nacelle

yaw, rotor furl, generator azimuth, tail furl accounts for four DOFs respectively.

The five flexible bodies are the three blades, tower and drive train. The blades

flexibility accounts for 1st-flapwise, 2nd-flapwise, and edgewise DOFs, so a total of three

DOFs are necessary to describe one blade. In the case of tower, two fore-aft, and two

side-to-side DOFs are accounted for, and the remaining one DOF is for drive train

flexibility. To sum up, 24 DOFs are required for one floating wind turbine with 3 blades,

and the DOFs will be further extended for MUFOWT.

To describe the kinematics and kinetics expressions of the wind turbine, several

reference frames formed by orthogonal sets of unit vectors are employed in FAST. The

major coordinate systems used for the FAST design code are listed in Table 4.1.

31

Table 4.1 Coordinate system of wind turbine

Unit Vector Set Description

z Inertial coordinates

a Tower base / Platform coordinate

t Tower element-fixed coordinate

b Tower top / base plate coordinate

d Nacelle / yaw coordinate

rf Rotor-furl coordinate

c Shaft coordinate

e Azimuth coordinate

f Teeter coordinate

g Hub / delta-3 coordinate

g’ Hub (prime) coordinate

i Coned coordinate

j Blade / pitched coordinate

Lj Blade coordinate system aligned with local structural axes

n Blade element-fixed coordinate

m Blade element-fixed coordinate for aerodynamics loads

te Trailing edge coordinate

tf Tail-furl coordinate

p Tail fin coordinate

Since a complete set of coordinate systems is defined, the transformation of fixed

quantities from one coordinate system to any other coordinate system is available.

Examples of simple transformation matrices used in FAST are shown below.

32

From tower base (Platform) to inertial:

1 6 6 4 4 1

2 5 5 4 4 2

3 6 6 5 5 3

cos 0 sin 1 0 0 cos sin 0

0 1 0 0 cos sin sin cos 0

sin 0 cos 0 sin cos 0 0 1

z q q q q a

z q q q q a

z q q q q a

(4.1)

where 4q , 5q , and 6q are roll, pitch and yaw angle of floating platform.

From tower top to tower base (Platform):

1 7 7 1

2 7 8 7 8 8 2

3 7 8 7 8 8 3

cos sin 0

sin cos cos cos sin

sin sin cos sin cos

a b

a b

a b

(4.2)

where 7 is longitudinal angle of tower top slope, 8 is lateral angle of tower top slope.

From nacelle yaw to tower top:

1 11 11 1

2 2

3 11 11 3

cos 0 sin

0 1 0

sin 0 cos

b q q d

b d

b q q d

(4.3)

where 11q is nacelle yaw angle.

From shaft tilt to nacelle yaw:

1 1

2 2

3 3

cos sin 0

sin cos 0

0 0 1

T T

T T

d c

d c

d c

(4.4)

where T is shaft tilt angle.

33

From azimuth to shaft tilt:

1 1

2 13 14 13 14 2

3 13 14 13 14 3

1 0 0

0 cos sin

0 sin cos

c e

c q q q q e

c q q q q e

(4.5)

where 13q is azimuth angle and 14q is zero azimuth offset due to the drive train flexibility.

Since the delta-3 angle and teeter angle for 3 bladed turbines are assumed to be

zero,

1 1 1

2 2 2

3 3 3

e f g

e f g

e f g

(4.6)

From blade-oriented hub to hub:

1 1

2 2

3 3

1 0 0

2 20 cos sin

2 20 sin cos

B B

B B

g g

g gN N

g g

N N

(4.7)

where BN is blade number. 1,2,3BN

From coning to blade-oriented hub:

1 1

2 2

3 3

cos 0 sin

0 1 0

sin 0 cos

g i

g i

g i

(4.8)

where is coning angle.

34

From blade pitch to coning:

1 1

2 2

3 3

cos sin 0

sin cos 0

0 0 1

P P

P P

i j

i j

i j

(4.9)

where P is blade pitch angle.

From blade twist to blade pitch

1 1

2 2

3 3

cos sin 0

sin cos 0

0 0 1

S S

S S

j Lj

j Lj

j Lj

(4.10)

where S is structural twist angle of blade.

There are 24 DOFs for three-bladed floating wind turbines, and each DOF is

tabulated in Table 4.2. All of the wind turbine motion can be described by those

variables.

Table 4.2 Degree of freedom variables

Variable Description

q1 Platform surge

q2 Platform sway

q3 Platform heave

q4 Platform roll

q5 Platform pitch

q6 Platform yaw

q7 Longitudinal tower top displacement for natural mode 1

q8 Latitudinal tower top displacement for natural mode 1

35

Table 4.2 Continued

Variable Description

q9 Longitudinal tower top displacement for natural mode 2

q10 Latitudinal tower top displacement for natural mode 2

q11 Nacelle yaw angle

q12 Rotor furl angle

q13 Generator azimuth angle

q14 Drive train rotational flexibility

q15 Tail furl angle

q16 Blade 1 flapwise tip displacement for natural mode 1

q17 Blade 1 edgewise tip displacement








Blades can be declined, or angled slightly downwind as denoted by the coning

angle . Similarly, each blade can be pitched or twisted independently so the

transformation matrices (4.8) ~ (4.10) can be used for any blade with each coning angle,

pitch angle, and twisted angle together with the reference frame specified for each blade.

36

4.3 Blade and Tower Flexibility

The flexibility of blades and towers in FAST is implemented by cantilevered

beams, fixed at one end to either the hub or the platform, and free at the other end. Both

have continuous distributed mass and stiffness, and the flexibility of structures is roughly

estimated by the normal mode shape summation method. By this simplification, the total

number of DOFs can be reduced from infinity to N , the number of dominant normal

modes, and the longitudinal or lateral deflection of the flexible beam can be expressed

by ,u z t as a function of distance z along the beam and time t .

1

,N

a aa

u z t z q t

(4.11)

where a z represents the normal mode shape and aq t denotes the generalized

coordinate.

In case of the tower deflection, both longitudinal and lateral directions are

represented by two modes which require two DOFs in each direction. Components of the

longitudinal and lateral displacement of the tower top consist of the contributions from

the first and second mode shapes. They are related to the tower DOFs as follows

7 7 9u q q (4.12)

8 8 10u q q (4.13)

where 7u is the total tower top fore-aft displacement, and 8u is the total tower top side-

to-side displacement. The tower top rotation angles can be also expressed as

7 7 7 9 9q q (4.14)

37

8 8 8 10 10q q (4.15)

where 7 is a rotation about 3a , 8 is a rotation about 1a . is the first derivative of the

mode shapes.

Blade deflection is modeled with two vibration modes for out-of-plane direction

and one mode for in-plane direction. This means that a total of three DOFs is required

for one blade. The current position of the local blade element can be expressed in root-

fixed coordinates as

1 2 3, , , ,z t u z t i v z t i r z w z t i u (4.16)

where the vector u is the vector position of the local blade element, 3i is along the blade,

1i is in the out-of-plane direction, 2i is in the in-plane direction, and r z is the distance

along the undeformed blade to the current blade element.

4.4 Kinematics

Once the geometry, coordinate system, and DOFs are set up, then the kinematic

expression of the wind turbine system can be derived. This step is necessary to build

kinetics expression, and develop Kane’s equations of motion. In this kinematics

expression, the vectors from any references frame can be transformed to a common

coordinate system and the acceleration of any point in the body can be expressed using

velocities and angular velocities. The velocities and angular velocities of one frame with

respect to another, say of frame B with respect to frame A , will be denoted by A Bv

and A B respectively.

38

The reference frames in wind turbine systems are presented in Figure 4.1(a).

Based on the given frames, the angular velocity of the tower base in the inertial frame

can be expressed by the summation of angular velocity of rotational motion of the

platform (roll, pitch and yaw) and is given by

4 1 5 2 6 3E X q q q z z z (4.17)

(a) (b) Figure 4.1 Turbine reference frames (a) and reference points (b)

X

F

B

N G

L

H

Z

T0

O

T

U P

S

Q

E (Earth)

39

The angular velocity of the tower top base plate in the inertial frame can be

expressed by the summation of angular velocity of longitudinal and lateral direction and

is given by

8 1 7 3X B a a (4.18)

The angular velocity of the nacelle relative to the tower-top base plate depends

on the rate of yaw

11 2B N q d (4.19)

The angular velocity of azimuth angle is

13 14 1N L q q + e (4.20)

Once the angular velocities of each reference frame are written, the angular

velocity of the low-speed shaft can be expressed in the inertial frame by combining

together.

4 1 5 2 6 3 8 1 7 3 11 2 13 14 1

E L E X X B B N N L

q q q q q q

z z z a a d + e

(4.21)

Since, there is no angular velocity difference between the low-speed shaft and

hub because of the absence of a teeter pin, the angular velocity of hub can be written as

E H E L (4.22)

Similarly, the velocity of the platform reference point (Z) depicted in Figure

4.1(b), which is dependent on the platform velocity in the inertial frame, is

1 1 2 2 3 3E Z q q q v z z z (4.23)

The velocity of the tower base in the inertial frame is

40

0 0E T E Z E X ZT v v r (4.24)

The velocity of the tower top base plate (O) in the tower base frame without axial

deflection is

07 1 8 3

E O E T u u v v a a (4.25)

where 7u is the tower top fore-aft deflection velocity, and 8u is the tower top side-to-

side deflection velocity. Those deflection velocities consider the axial-reduction terms.

The axial reduction of the tower is the combined result of assuming the flexible beam

with a fixed length and the fact that the free end of a cantilever beam must move closer

to the fixed end when the beam deflects laterally. A detailed derivation of tower

deflection in Figure 4.2 is presented by J. Jonkman. (Jonkman, 2003)

Figure 4.2 Tower deflection geometry (Jonkman, 2003)

The velocity of point T on the flexible tower in the inertial frame is

41

07 1 9 2 1 8 1 10 2 3

E T E TT T T Tq h q h q h q h v v + a + a (4.26)

where h is the elevation of point T which ranges from zero to H . The elevation h

equals zero at the top of the rigid part of the tower.

The velocity of the nacelle center of mass (U) in the inertial frame is

E U E O E N OU v v r (4.27)

where E N is the angular velocity of the nacelle in the inertial frame

E N E X X B B N , and OUr is the position vector pointing from the tower-top

base plate to the nacelle center of mass.

The velocity of the hub in the inertial frame is

E P E O E N OP v v r (4.28)

Similarly, the velocities of the blade axes intersection point (Q) and the hub

center of mass in the inertial frame are

E Q E P E H PQ v v r (4.29)

and

E C E Q E H QC v v r (4.30)

Finally, the velocity of any point S on blade 1 in the inertial frame is

E S E Q H S E H QS v v v r (4.31)

where H Sv is the velocity of point S on blade 1 with respect to the rotating frame fixed in

the hub, and QSr is the position vector connecting any point S on the deflected blade 1 to

the blade axes intersection point Q that can be expressed as

1 2 3, , ,QSHu r t v r t r R w r t r i i i (4.32)

42

The H Sv can be obtained by time derivative of QSr

1 2 3, , ,H S u r t v r t w r t v i i i (4.33)

So, the velocity of any point S on blade 1 in the inertial frame can be written as a

summation of the derived equation above

01 1 2 2 3 3 4 1 5 2 6 3

7 9 1 8 10 3 4 1 5 2 6 3 8 1 7 3 11 2

4 1 5 2 6 3 8 1 7 3 11 2 13 14 1

4 1 5 2 6 3 8 1 7 3 11 2

[ + ]

[

E S ZT

OP

PQ

q q q q q q

q q q q q q q q

q q q q q q

q q q q

v z z z z z z r

a a z z z a a d r

z z z a a d e r

z z z a a d

13 14 1] QS H Sq +q e r v

(4.34)

This velocity can be simply expressed as a generalized form

24

1

E A E A E Ar r t

r

q

v v v (4.35)

where E Arv is the r th

partial velocity associated with point A, and E Atv is the sum of all

other terms. The angular velocity of any reference frame A in the inertial frame can also

be expressed as

24

1

E A E A E Ar r t

r

q

(4.36)

where E Ar is the r th

partial angular velocity associated with point A, and E At is the sum

of all other terms.

The acceleration of any point in the inertial frame can be derived by time

derivatives of the E Av as

24 24

1 1

E A E A E A E A E Ar r r r t

r r

d d dq q

dt dt dt

a v v v v (4.37)

43

or it can be expressed in a similar form as the velocity form:

24

1

E A E A E Ar r t

r

q

a v a (4.38)

where,

24

1

E A E A E At r r t

r

d dq

dt dt

a v v (4.39)

4.5 Generalized Active Forces

Using partial velocity vectors derived above as base vectors (direction vectors), it

is useful to project forces along these vectors. Those projected forces are called

generalized forces. Consider a set of forces iXF 1,2,...,i N applied at points iX ,

then the generalized forces are obtained by adding the generalized forces from the

individual forces as

1

i i

NX XE

r ri

F F

v

1,2,...,r n (4.40)

where N is the number of rigid bodies in the system, and n is the number of DOFs.

We can obtain the kinematic expressions of points iX using the velocity at the

center of mass of rigid body G as

i iX GXE E G E G v v r (4.41)

where, E G is the angular velocity associated with the rigid body, and iGXr is the

position vector of the center of mass of the rigid body relative to the center of mass G .

44

By differentiating Equation (4.41) with respect to the generalized coordinate r ,

then

i iX GXE E G E Gr r r v v r

1,2,...,r n (4.42)

Substituting Equation (4.42) into (4.40) gives

1

1 1

1 1

i

i

i

NGXE G E G

r r r ii

N NGXE G E G

r i r ii i

N NGXE G E G

r i r ii i

F F

F F

F F

v r

v r

v r

(4.43)

We can rewrite this in the form

E G E Gr r rF v F T (4.44)

where F and T are the resultant force and moment induced by the set of applied forces

iXF , respectively.

The all resultant forces and moments acting on elements of the floating wind

turbine contribute to the total generalized active forces. These forces include the

hydrodynamic forces, mooring restoring forces, aerodynamic forces, elastic forces from

the tower, blades, and drive train flexibility, elastic forces from the nacelle yaw spring,

gravitational forces, generator forces, and damping forces.

r r r r r r r rHydro Mooring Aero Elastic Gravity Generator DampingF F F F F F F F (4.45)

The detailed derivations of all generalized active forces are beyond the scope of

this study.

45

4.6 Generalized Inertia Forces

Along with the generalized active forces, which are presented above, generalized

inertia forces can be also defined. Similarly, a generalized inertia force is defined as a

projection of inertia force along a partial velocity vector.

*

1

i i

NX XE E

r r ii

F m

v a

1,2,...,r n (4.46)

where, as before, *rF is the inertia force on points iX and iXE

rv is the rth partial velocity

associated with rigid body, iXE a is the acceleration of points iX . Using Equation (4.42),

the partial velocity iXErv can be replaced as

*

1

1

1 1

1 1

* *

i i

i i i

i i i

i i i

NGX XE G E G E

r r r ii

NX GX XE G E E G E

r i r ii

N NX GX XE G E E G E

r i r ii i

N NX GX XE G E E G E

r i r ii i

E G E Gr r

F m

m m

m m

m m

v r a

v a r a

v a r a

v a r a

v F T

(4.47)

where *F and *T are force and moment passing through the center of mass respectively

and written as

* GM F a (4.48)

* T I I (4.49)

46

where M is the mass of rigid body, I is the central inertia dyadic of the rigid body, Ga is

the acceleration of center of mass, and are the angular velocity and angular

acceleration of the rigid body respectively.

For the floating wind turbine model in FAST, the generalized inertia forces are

separately calculated from the rigid body components with mass, then summed up

together to get the total generalized inertia forces.

* * * * * *r r r r r rPlatform Tower Nacelle Hub Blades

F F F F F F (4.50)

The detail derivations of all generalized inertia forces are beyond the scope of

this study so are not presented.

4.7 Kane’s Equations

FAST uses Kane’s method to derive the dynamic equations of motion. Once the

equation of motion is set up, it can be solved by numerical integration. Kane’s method

provides an elegant formulation of the dynamical equations of motion, and it simply

state that the sum of the generalized active forces and the generalized inertia forces, for

each generalized coordinate, is zero. That is

* 0r rF F

1,2,...,r n (4.51)

Kane first published the above equation in 1961 (Kane, 1961). Intuitively, the

Kane’s equations can be interpreted as the sum of a projection of the applied and inertia

forces along the directions of the partial velocity vectors is zero.

47

By substituting the kinematic expressions in the previous section into Equation

(4.51), we can obtain a set of n coupled dynamic equations of motion of the entire

floating wind turbine system, which can be written as

1

, 0n

rs s r r rs

C q f q q

1,2,...,r n (4.52)

The first term in Equation (4.52) includes the known coefficient rsC , and the

acceleration terms, while the second term includes the lower order terms as a function of

velocity and displacement of each degree of freedom. This equation is expanded into

matrix form as

11 12 1 1 1 1 1

21 22 2 2 2 2 2

1 2

( , )

( , )

( , )

( , )

n

n

rs s r r r

n n nn n n n n

C C C q f q q

C C C q f q q

C q f q q

C C C q f q q

(4.53)

At each time step, the right hand side of Equation (4.53) is filled up using the

fourth-order Adams-Bashforth predictor method; the Gauss elimination technique is then

applied to obtain the accelerations of the entire DOFs. These accelerations are then used

as the estimate for the next step. Finally, a fourth-order Adams-Mounton corrector is

used to make the final estimate and to determine the accelerations. Once the motion and

applied load are specified by solving the equations of motion, then the local loads at

various point of the wind turbine can be calculated by performing simple summations of

these loads.

48

4.8 Coupling between FAST and CHARM3D

CHARM3D is 3D hull-mooring-riser fully coupled static/dynamic analysis tool

which was developed by Prof. M.H. Kim’s research group during the past two decades

(Kim et al., 2001; Ran et al., 1999; Tahar and Kim, 2003; Yang and Kim, 2010). The

CHARM3D has been verified through numerous comparisons against experiments and

field data during the past decade. Figure 4.3 shows a simple example that CHAMR3D

can handle. The hydrodynamic coefficients including added mass, radiation damping,

first and second order wave forces and mean drift forces of floater are obtained by a 3D

diffraction/radiation preprocessor WAMIT in the frequency domain, and then transferred

to CHARM3D for the time domain calculation.

Figure 4.3 Coupled hull and mooring, riser analysis

Viscous loading

1st order wave loading 2nd order wave loading

Wind loading

Current loading

Hydrodynamic loading on

mooring lines

Mooring line inertia & drag

Hull-mooring line coupled dynamics

Seabed interactions

49

In CHARM3D, the floating platform is assumed to be a rigid body undergoing

motion in waves, winds, and currents, and the mooring lines are modeled as a higher-

order finite element model. The mooring line dynamics due to the wave kinematics and

its inertia and drag can be also included in the analysis. The entire mooring/riser

dynamics and hull motions are solved simultaneously in a combined system matrix at

each time step.

The equation of motion of a floating body in the time domain can be expressed as

below.

[ ( )] ( ) ( , ) ( , ) ( , ) ( )aI c n m hsM M t t t t F F F F F (4.54)

where, ( )I tF is wave exciting force, ( , )c t F is radiation damping force, ( , )n t F is

nonlinear viscous drag force, ( , )m t F is mooring restoring force and ( )hs F is

hydrostatic force. M and ( )aM are the floating body mass and added mass at infinite

frequency respectively.

At the initial stage of coupling, the platform added mass at infinite frequency,

( )aM and hydrostatic coefficients are transferred to FAST from CHARM3D. During

the time marching simulation, FAST calculates all the dynamics of blade, rotor, tower

including floating platform, and CHARM3D feeds the required forces of the platform

into FAST. Those forces are properly fed into the generalized active forces in Equation

(4.45), and the platform added mass transferred to FAST is used to calculate the

generalized inertia force of the platform inside the FAST. The forces calculated by

CHARM3D include the hydrodynamic wave force (first-order wave-frequency and

50

second-order sum- and difference-frequency), viscous force of Morison members,

radiation damping force in the form of convolution integral, and mooring restoring force.

The mooring restoring force can be estimated from the top tension of each mooring line

and its directional cosine. The nonlinear viscous drag forces on Morison members are

evaluated at the instantaneous body position and up to the instantaneous free-surface

elevation. Then FAST computes the entire dynamics of wind turbine including platform

response using the forces from CHARM3D and returns the updated displacement and

velocity of platform to CHARM3D. Then CHARM3D recursively calculates the updated

forces based on the new position and velocity of platform. Figure 4.4 shows the

schematic diagram of coupling between CHARM3D and FAST. A possibly simpler

coupling approach through transmitted forces and moments at the tower base was also

experimented by Shim and Kim (Shim and Kim, 2008) and it was seen that it cannot

fully account for the entire features of the more complicated couplings between the rotor

and floater.

The time step of the CHARM3D side should be determined by the time step of

the FAST side since FAST requires platform forces, which are generated by CHARM3D,

at every internal time step. So, both modules should have the similar time step or

CHARM3D can have a slightly larger time step. The time step of the CHARM3D side

should be carefully determined so that CHARM3D may not feed large variation of

mooring restoring force to FAST.

51

Figure 4.4 Data transfer between CHARM3D and FAST

C

H

A

R

M

3

D

F

A

S

T

Wave Exciting Force

Radiation Damping Force

Viscous Drag Force

Mooring Restoring Force

Added Mass & Hydrostatic Coefficient

Platform Displacement

Platform Velocity

52

5. DYNAMICS OF A MULTIPLE UNIT FLOATING OFFSHORE WIND

TURBINE

5.1 Introduction

In this chapter, the dynamic equation of motion for multiple unit offshore wind

turbines in the time domain is established by utilizing the equations of motion for single

floating wind turbines. The dynamic behavior of MUFOWT can be derived from full

DOFs of single floating wind turbines including 6-DOFs of floating platform, and

additional wind turbine DOFs with proper platform-turbine coupling terms. The entire

MUFOWT system equations of motion are built in one global coefficient matrix with

forcing functions, and then solved simultaneously at each time step. Assuming that every

degree of freedom for a three-bladed turbine in FAST is turned on, the total DOFs of

MUFOWT can be expressed as 6 18 N , where N is total number of turbines. The

generalized inertia and active forces from each turbine should be independently taken

into account and applied to the floating platform at the same time, but the generalized

inertia and active force of a floating platform should be included only once. The coupled

terms between a floating platform and each turbine in the coefficient matrix should be

derived by accounting for every effect of generalized inertia and active forces from both

bodies. In the case of the coupling terms between one turbine and another one, the

coefficients are set to zero because a direct kinematic or kinetic relationships between

each turbine does not exist.

53

5.2 Equations of Motion

The equations of motion for MUFOWT can be basically derived by summation

of equations of motion from the single wind turbine. Assuming that there are total N

turbines in one floating platform as can be seen in Figure 5.1, the total generalized active

forces and the generalized inertia forces of N turbines and the floating platform are now

derived.

Figure 5.1 Schematic configuration of MUFOWT

5.2.1 Generalized Active Force

The generalized active force on the whole MUFOWT system can be divided into

turbine part and floating platform part. For turbines on top of the platform, the

generalized active force can be obtained separately from each turbine. Aero dynamic,

elastic, gravity, generator, and damping force for each turbine are summed up

Floating Platform

Turbine #1 Turbine #2

Turbine #N ….

….

54

individually and arranged one by one. Then the generalized active force of each turbine

without floating platform can be expressed as

#1 #1 #1 #1 #1#1

#2 #2 #2 #2#2

Turbine Turbine Turbine Turbine TurbineTurbine

r r r r r rAero Elastic Gravity Generator Damping

Turbine Turbine Turbine TurbineTurbiner r r r r rAero Elastic Gravity Generator D

F F F F F F

F F F F F F

#2

# # # # ##

Turbine

amping

Turbine N Turbine N Turbine N Turbine N Turbine NTurbine Nr r r r r rAero Elastic Gravity Generator Damping

F F F F F F

(5.1)

So, the total generalized active forces of N turbines excluding floating platform

can be written as a summation of Equation (5.1).

#1 #2 # Total Turbines Turbine Turbine Turbine Nr r r rF F F F (5.2)

The generalized active force on the floating platform should be taken into

account separately from the turbines because the platform is not individually arranged

but positioned as a whole body and expressed as

Platformr r rHydro Mooring

F F F (5.3)

This means that the generalized active force on the platform comes from

hydrodynamic force and mooring line restoring force, and should be accounted for only

once.

By combining Equations (5.2) and (5.3), the total generalized active forces on the

MUFOWT can be established as shown below in Equation (5.4).

55

#1 #2 #

#1 #1 #1 #1 #1

#2

Total Turbine Platform Turbine Turbine Turbine N Platformr r r r r r r

Turbine Turbine Turbine Turbine Turbine

r r r r rAero Elastic Gravity Generator Damping

Turbine

r rAero Elastic

F F F F F F F

F F F F F

F F

#2 #2 #2 #2

# # # # #

+

+

Turbine Turbine Turbine Turbine

r r rGravity Generator Damping

Turbine N Turbine N Turbine N Turbine N Turbine N

r r r r rAero Elastic Gravity Generator Damping

r rHydro Mooring

F F F

F F F F F

F F

(5.4)

5.2.2 Generalized Inertia Force

Similarly, the generalized inertia force can be also expressed for MUFOWT.

First, the generalized inertia forces from each turbine are obtained. The inertial loadings

of each turbine, such as tower, nacelle, hub and blades are calculated based on the mass

and inertia properties of each component and summarized with respect to the tower base

origin for each turbine. The generalized inertia force of each turbine can be expressed as

#1 #1 #1 #1* #1 * * * *

#2 #2 #2 #2* #2 * * * *

* # *

Turbine Turbine Turbine TurbineTurbiner r r r rTower Nacelle Hub Blades

Turbine Turbine Turbine TurbineTurbiner r r r rTower Nacelle Hub Blades

Turbine Nr r Tower

F F F F F

F F F F F

F F

# # # #* * *Turbine N Turbine N Turbine N Turbine N

r r rNacelle Hub BladesF F F

(5.5)

Thus, total generalized inertia forces of turbines without floating platform are

* * #1 * #2 * #Total Turbine Turbine Turbine Turbine Nr r r rF F F F (5.6)

If we include the generalized inertia force from the floating platform obtained by

the platform mass property, then the total generalized inertia force of MUFOWT can be

expressed as below.

56

* * * * #1 * #2 * # *

#1 #1 #1 #1* * * *

#2 #2* *

Total Turbine Turbine Turbine Turbine Nr r r r r r rPlatform Platform

Turbine Turbine Turbine Turbine

r r r rTower Nacelle Hub Blades

Turbine Turbine

r rTower Nacelle

F F F F F F F

F F F F

F F

#2 #2* *

# # # #* * * *

*

+

Turbine Turbine

r rHub Blades

Turbine N Turbine N Turbine N Turbine N

r r r rTower Nacelle Hub Blades

r Platform

F F

F F F F

F

(5.7)

Note that the inertia force from the platform is only accounted for once to obtain

the total generalized inertia force of MUFOWT system.

The velocity vectors inside the above generalized forces should be calculated

based on the tower base position vectors which are uniquely determined by the location

of each turbine base relative to the platform reference point.

0 0

# #

E T E Z E X ZT

Turbine N Turbine N v v r (5.8)

Figure 5.2 Tower base position vectors

Turbine #1 Turbine #2 Turbine #N ….

57

For example, the relative velocity vector of tower base position in Equation (5.8)

is determined by a different tower base position vector of 0ZTr in Figure 5.2.

Once the tower base velocity, acceleration, and partial velocity vectors for each

turbine are derived based on the different tower base position vector above, the

consecutive kinematic vectors such as tower top, nacelle, hub and blade vectors are

determined successively. Finally, the total equations of motion of MUFOWT can be

established using Kane’s equation as we did in the single turbine analysis.

5.3 Coefficient Matrix of Kane’s Dynamics

In case of a three-bladed single floating turbine, the coefficient matrix rsC in

Equation (4.53) can be expressed by components from the floating platform to the blades.

Figure 5.3 Coefficient matrix of single turbine and platform

Blade #1

Platform

Tower Flexibility

Nacelle & Drivetrain

Blade #2

Blade #3

(a) (b) (c)

(d) (e)

(f)

(g)

58

The coefficient matrix in Figure 5.3 shows that the dynamic responses of turbine

part, including tower, nacelle and blades, are coupled with the platform DOFs as there

are off-diagonal terms between platform and turbine which are represented by (a) ~ (c)

terms. The off-diagonal terms denoted by (d) ~ (g) terms represent the dynamic coupling

between turbine components. For example, the terms (d) represent the coupling between

tower flexibility and the nacelle dynamics, and (f) denote the coupling between nacelle

or drivetrain and blades. It is seen that there is no dynamic coupling between each blade,

so the off-diagonal terms (g) are null matrix. This is true because there is no direct

kinematic or kinetic relationship between each blade as their dynamics can be

transmitted to another blade only through the hub.

Figure 5.4 Coefficient matrix of multiple turbines and platform

Platform (A)

Turbine #1

Turbine #2

Turbine #N

…(B) (C)

(E)

(D)

(F)

(G)

59

In the case of MUFOWT, it is expected that the dynamic coupling terms between

one turbine and another do not exist since the turbine response of one turbine can affect

another turbine only through the floating platform. In this regard, the dynamic coupling

terms between the floating platform and each turbine are important and should be

properly taken into consideration. So, the coefficient matrix rsC for MUFOWT can be

depicted as in Figure 5.4.

In this global coefficient matrix, the off-diagonal terms (B) and (C) represent the

dynamic coupling between the floating platform and turbine #1 and turbine #2

respectively. As mentioned above, the off-diagonal terms (E) ~ (G), which represent the

dynamic coupling between one turbine and another, are null matrix. The only way to

interact between two turbines is through the floating platform which is represented in

terms (B) ~ (D). If there are connecting structural members between two separated

turbine towers, such as brace or truss, then the coupling terms between the two turbines

(E) ~ (G) should not be zero.

As pointed out earlier, the coefficient matrixes for each turbine are determined

based on the different tower base position vectors as well as the dependent platform

coupling terms (B) ~ (D).

5.4 Force Vector of Kane’s Dynamics

The right hand side of Equation (4.53) would be forcing terms as a function of

displacement and velocity of each degree of freedom. The forcing function represented

by ,r r rf q q for a single floating turbine is depicted in Figure 5.5.

60

Figure 5.5 Forcing vector of single turbine and platform where, (a) is the platform loading, which includes the loadings from the turbine part, (b)

is the tower top fore-aft or side-to-side loading, (c) is the nacelle and drivetrain loading,

(d) ~ (f) represent the blade loadings. In case of a floating wind turbine, the

hydrodynamic loading due to radiation damping, hydrodynamic wave loading,

hydrostatic loading, viscous loading, and mooring line restoring loading should be

included in these (a) terms. Hydrodynamic added mass effect is not included in this part,

but included in the coefficient matrix in the previous section. Aerodynamic loading on

the blades should be positioned in (d) ~ (f). In a similar way, the force vector for

MUFOWT can be established.

(a)

(b)

(c)

(d)

(e)

(f)

61

Figure 5.6 Forcing vector of multiple turbines and platform

The forcing term (A) in Figure 5.6 includes the hydrodynamic, hydrostatic

loading, viscous loading, mooring line restoring loading, and the loading from each

turbine. Additional loading terms on each turbine should be followed. Those turbine

loading terms are also uniquely determined based on the position of the tower base.

5.5 Equations of Motion in FAST

The floating wind turbine solver FAST is basically designed to analyze a single

floating wind turbine. Thus, the maximum DOFs for a three-bladed turbine are limited to

24. In this study, all of the global variables inside the FAST, including the AeroDyn

module, are expanded so that it can afford one more dimension. For example, single

variables are converted to the one dimensional array, and n dimensional variables are

expanded to 1n dimensional variables. One additional space is allocated for turbine ID

Turbine #N

…

(A)

Turbine #1

Turbine #2

62

so all the inside calculations can be done for different turbines. One of the challenges in

this work is to combine individual equations of motion for each turbine into global

equations of motion. Since FAST is designed to build a full coefficient matrix rsC for a

single turbine, rigorous modification work has been done in order to build a global

matrix and solve the entire MUFOWT system simultaneously at given time step.

Figure 5.7 Series of coefficient matrices As can be seen in Figure 5.7, FAST starts to build the coefficient matrix for

turbine #1 including the dynamic effect of floating platform, then repeat building the

next coefficient matrix for turbine #2. The first step for filling up the coefficient matrix

for turbine #1 is quite a normal procedure, but attention should be paid for the next step

to establish the coefficient matrix for turbine #2 through turbine #N.

Since the inertia terms of the floating platform are already taken into account

when the first coefficient matrix is built in (a), the platform inertia due to its mass and

added mass should be eliminated from the (c) and (e) terms. This does not mean that

those terms are null matrix because those portions still include the inertia loadings from

the turbine. So, the coefficient matrix of (a), (c) and (e) can be expressed as

(a)

Turbine #1

Turbine #2

Turbine #N

…

(b) (c) (d) (e) (f)

(b) (d) (f)

63

( ) #1 #1 #1

#1 #1

rs rs rs rs rs rsa Platform Hydro Tower Nacelle Rotor

rs rsHub Blades

C C C C C C

C C

(5.9)

( ) #2 #2 #2 #2 #2rs rs rs rs rs rsc Tower Nacelle Rotor Hub BladesC C C C C C (5.10)

( ) # # # # #rs rs rs rs rs rse Tower N Nacelle N Rotor N Hub N Blades NC C C C C C (5.11)

On the other hand, the dynamic coupling terms between platform and turbines,

such as (b), (d) and (f) should contain the platform inertia terms including hydro added

mass term because the platform inertia has to influence all turbines.

( ) #1 #1 #1

#1 #1

rs rs rs rs rsb Platform Tower Nacelle Rotor

rs rsHub Blades

C C C C C

C C

(5.12)

( ) #2 #2 #2

#2 #2

rs rs rs rs rsd Platform Tower Nacelle Rotor

rs rsHub Blades

C C C C C

C C

(5.13)

( ) # # #

# #

rs rs rs rs rsf Platform Tower N Nacelle N Rotor N

rs rsHub N Blades N

C C C C C

C C

(5.14)

Finally, the global coefficient matrix with N turbines can be made by combining

every coefficient matrix as a formation of Figure 5.4. Turbine inertia terms, (c) and (e),

should be added to the platform inertia matrix in turbine #1 represented by (A) in Figure

5.4. The other terms like (B) through (D) in Figure 5.4 are assigned by (b) ~ (f) as

expressed below so that the global matrix has 6 platform DOFs plus 18 N turbines

DOFs.

( ) ( ) ( ) ( )rs rs rs rsA a c eC C C C (5.15)

64

( ) ( )rs rsB bC C (5.16)

( ) ( )rs rsC dC C (5.17)

( ) ( )rs rsD fC C (5.18)

The global force vector can also be made in a similar way. As can be seen in

Figure 5.8, each force vector is built one by one and then combined so that it forms a

global force vector.

Figure 5.8 Series of forcing vectors The platform loading terms in the turbine #2 vector through the turbine #N vector,

represented by (b) and (c) terms in Figure 5.8, should not include the platform loadings

such as hydrodynamic, hydrostatic, viscous, and mooring restoring forces. Those

loadings are already included in the force vector in (a). Similarly, the terms (b) and (c)

still contain the turbine loading portions.

( ) #1 #1

#1 #1 #1

, , , , ,

, , ,

r r r r r r r r r r r r r r ra Platform Hydro Tower Nacelle

r r r r r r r r rHub Blades Aero

f q q f q q f q q f q q f q q

f q q f q q f q q

(5.19)

(a)

Turbine #1

… (b)

Turbine #2

Turbine #N

(c)

65

( ) #2 #2 #2

#2 #2

, , , ,

, ,

r r r r r r r r r r r rb Tower Nacelle Hub

r r r r r rBlades Aero

f q q f q q f q q f q q

f q q f q q

(5.20)

( ) # # #

# #

, , , ,

, ,

r r r r r r r r r r r rc Tower N Nacelle N Hub N

r r r r r rBlades N Aero N

f q q f q q f q q f q q

f q q f q q

(5.21)

Finally, the global force vector in Figure 5.6 can be made by summation of the

platform loading terms, and distributing all the other turbine force vectors successively.

The forcing term (A) in Figure 5.6 is going to be

( ) ( ) ( ) ( )

, , , ,r r r r r r r r r r r rA a b cf q q f q q f q q f q q (5.22)

Once the global coefficient matrix and force vector are established, then the

built-in Gauss elimination solver inside the FAST can handle those global equations of

motion for MUFOWT.

5.6 Consideration of the Shade Effect

The interference or shade effect of turbines in the MUFOWT is one of the

biggest concerns in designing the size and capacity of a platform. Figure 5.9 shows the

considerable wake effect observed behind the Horns Rev offshore wind farm west of

Denmark. Due to this wake effect between turbines, the efficiency of the power

production will be significantly decreased and the dynamic loadings of downstream

turbines will be increased.

66

Figure 5.9 Wake turbulence behind individual wind turbines (Courtesy of Vattenfall Wind Power, Denmark)

To minimize this wake effect, turbines are typically arranged with enough

spacing. The distance between wind turbines is commonly determined by the rotor

diameter and local wind conditions. One of the research suggests spacing turbines

between 5 and 10 rotor diameters apart. If prevailing winds are generally from the same

direction, turbines may be installed 3 or 4 rotor diameters apart in the direction

perpendicular to the prevailing winds. Under multidirectional wind conditions, a space

from 5 to 7 rotor diameters is recommended (Global Energy Concepts and AWS

Truewind LLC, 2005) between each turbine as can be seen in Figure 5.10.

67

Figure 5.10 Turbine spacing recommendation (Courtesy of http://en.openei.org/) It is known that the power loss of a downstream turbine can reach up to 40% in

full wake conditions; if various wind directions are considered the overall loss of power

is around 8% for onshore wind farms and 12% for offshore farms (Barthelmie et al.,

2009; Barthelmie et al., 2008). Hence, the accurate simulation of the wake effect

between neighboring turbines on a wind farm is very important. Recently, the

performances of several numerical wake models for offshore wind farm design have

been proposed and evaluated (Rados et al., 2001). More recently, the numerical

computation using CFD program with Actuator Line Method and Actuator Disc Method

are performed (Ivanell et al., 2007; Mikkelsen, 2003). However, the wake effects and

how they impact wind turbines and plant performance have not been well understood

due to the complex behavior of turbulent wind field at downstream side.

68

That shade effect could be a big concern in the MUFOWT design as well, since

the size of a platform is limited compared to the land site, and some of the rotors could

be partly located at a shadow area of the other turbines. This aspect has been addressed

by the CFD analysis and the model test was conducted by WindSea AS in Norway. As

introduced in the chapter 1, their semi-submersible platform supports two upwind

turbines and one downwind turbine (3.6MW each). They proved that the turbulent wind

field at the aft turbine is moderated and smaller than the one in land-based wind turbines.

Figure 5.11 Power production for downwind turbine (Lefranc and Torud, 2011)

According to their research (Lefranc and Torud, 2011), the loss of a power

production due to the wake effect is significant for the low wind velocity range while for

high velocity condition, the power production does not make big differences, as can be

seen in Figure 5.11. The upper line represents the power production of the downwind

turbine without any shade effect, while the lower line is for the downwind turbine

accounting the shade effect from the two upwind turbines. Risø calculated the annual

69

reduction ratio of a power production for the shaded turbine, and it was estimated at 25%.

If the front turbines are included in the efficiency estimation, then the reduction ratio

with a downtime of 15% is estimated to be 7%. This reduction factor is strongly

dependent on turbine properties, turbine position and given environmental conditions

thus an optimal arrangement of turbines and a proper site installation may reduce those

negative shade effects.

If the power reduction ratio and the increased turbulence intensity of the shaded

turbine are known parameters from the experiment, the wake effect can be partially

included in the numerical simulations by applying different wind field input for each

turbine. For example, the wake effect on the rear turbine in Figure 5.12 can be

numerically simulated with reduced wind velocity and increased turbulence intensity of

the separated rear-side wind field.

(a) (b) Figure 5.12 Global wind field (a) and separated wind field (b)

Separated wind field

70

In this study, numerical wake effects between neighboring turbines are not

assessed because of the uncertainty and the complexity. Instead, the arrangement of

turbines is determined so as to maximize the exposed area of each turbine and minimize

the overlapped areas.

71

6. CASE STUDY I: SINGLE-TURBINE HYWIND SPAR*

6.1 Introduction

Recently, considerable research progress has been made in the single turbine

floating platform area. Several different kinds of floating platforms are suggested as a

turbine base platform and their performance and cost effectiveness are checked

(Butterfield et al., 2007). Most general types of floating platforms for single wind

turbines are classified into three categories based on the physical properties that are used

to ensure static stability. The first type is a Spar-buoy type platform. This type of

platform achieves stability by ballast weights hung below a central buoyancy tank which

make a very low center of gravity for the whole system. The second type is a tension leg

platform (TLP) type which achieves stability through mooring line tension. The last type

is a barge or semi-submersible type. Stability in this type is achieved by the distributed

buoyancy and righting arm that may generate positive restoring moment for the platform.

Recently, it is regarded that the semi-submersible platform has become more popular

because of better response characteristics in an ocean environment compared to a barge.

In this chapter, the performances of a single turbine platform are assessed before

we move on to multiple turbine case studies. The Hywind spar type designed for 320m

water depth is selected as a floating platform and NREL’s 5MW baseline turbine is

mounted on top of the platform (Jonkman, 2010). The present analysis method integrates * Part of this chapter is reprinted with permission from “Aero-elastic-control-floater-mooring coupled dynamic analysis of floating offshore wind turbines” and “Influence of control strategy to FOWT hull motions by aero-elastic-control-floater-mooring coupled dynamic analysis” by Bae and Kim, 2011. Proceedings of the 21st (2011) International Offshore and Polar Engineering Conference, Copyright [2011] by the International Society of Offshore and Polar Engineers (ISOPE)

72

rotor dynamics and control, aero-dynamics, tower elasticity, floater dynamics, and

mooring-line dynamics to investigate the full dynamic coupling among them in the time

domain. The corresponding rotor-floater-mooring coupled dynamic analysis computer

program is developed by combining the respective modules. For the dynamics and

control of blade and tower, the primary design code of wind turbines, FAST, developed

by the National Renewable Energy Laboratory (NREL), is employed (Jonkman and Buhl

Jr, 2005). The portion of the FAST algorithm is modified to include some features of the

floater-mooring coupled dynamic analysis program, CHARM3D, and vice versa so that

the full coupling of rotor and floater can be accurately achieved.

The work presented in this chapter is based on the two conference proceedings

(Bae et al., 2011a; Bae et al., 2011b) presented in the 21st International Offshore and

Polar Engineering Conference.

6.2 Numerical Model for 5MW Hywind Spar

The adopted model of the 5MW turbine is the NREL offshore 5MW baseline

wind turbine which has been adopted as the reference model for the integrated European

UpWind research program. The Hywind floating platform in this case study is the ‘OC3-

Hywind’ spar-buoy type platform which is slightly different from the actual turbine used

by Statoil of Norway. The detailed specifications of 5MW turbine and Hywind spar hull

are summarized in Tables 6.1 and 6.2. The characteristics of the mooring system are

tabulated in Table 6.3.

73

Table 6.1 Specification of 5MW turbine

Item Unit Value

Tower height m 90.0

Rotor diameter m 126.0

Tower diameter (top) m 3.87

Tower diameter (bottom) m 6.5

Elevation to Tower Base above SWL m 10

Elevation to Tower Top above SWL m 87.6

Overall Tower mass kg 249,718

Total wind turbine weight (except for platform) kg 599,718

CM Location of Tower above SWL m 43.4

Tower Structural Damping Ratio (All modes) % 1

Table 6.2 Specification of Hywind spar platform

Item Unit Value

Depth to Platform Base below SWL m 120.0

Elevation to Platform Top Above SWL m 10

Depth to Top of Taper Below SWL m 4

Depth to Bottom of Taper Below SWL m 12

Platform Diameter Above Taper m 6.5

Platform Diameter Below Taper m 9.4

Platform Mass, including Ballast kg 7,466,330

CM Location Below SWL m 89.9155

Platform Roll Inertia about CM kg·m2 4,229,230,000

Platform Pitch Inertia about CM kg·m2 4,229,230,000

Platform Yaw Inertia about Platform Centerline kg·m2 164,230,000

74

The Hywind spar is moored by three catenary lines. To increase the yaw stiffness

of the platform, the lines are attached to the hull via a delta connection. This delta-

connection effect is included in the time domain simulation by adding the corresponding

yaw spring stiffness.

Table 6.3 Specification of Hywind spar mooring system

Item Unit Value

Number of Mooring Lines ea 3

Angle Between Adjacent Lines deg 120

Depth to Anchors Below SWL (Water Depth) m 320

Depth to Fairleads Below SWL m 70.0

Radius to Anchors from Platform Centerline m 853.87

Radius to Fairleads from Platform Centerline m 5.2

Unstretched Mooring Line Length m 902.2

Mooring Line Diameter m 0.09

Equivalent Mooring Line Mass Density kg/m 77.7066

Equivalent Mooring Line Weight in Water N/m 698.094

Equivalent Mooring Line Extensional Stiffness N 384,243,000

Additional Yaw Spring Stiffness Nm/rad 98,340,000

Each mooring line is modeled by 20 higher-order finite elements with an

unstretched length of 902.2m. Illustrations of mooring line arrangement are shown in

Figure 6.1.

75

Figure 6.1 Hywind spar mooring line arrangement 6.3 Hydrodynamic Coefficients in the Frequency Domain

Wave forces and hydrodynamic coefficients for the submerged portion of the hull

are calculated by using the potential-based 3D diffraction/radiation panel program (Lee

et al., 1991). The submerged body has two planes of symmetry and each quadrant has

3,900 panels as can be seen in Figure 6.2.

Figure 6.2 Discretized panel model of floating body (Hywind spar)

-20

0

-4-2024-4-2024

-120

-100

-80

-60

-40

-20

0

xy

z

76

Second-order mean drift forces are also calculated to generate slowly-varying

drift forces and motions through Newman’s approximation method. The viscous drag

force of the hull is included by employing two Morison members for the upper and

lower sections. The drag coefficient CD is taken to be 0.6 which is typical for a cylinder

at high Reynolds numbers.

The viscous loadings on Morison members are calculated at the body’s

instantaneous position up to the instantaneous free surface at each time step. The wave

particle kinematics above MWL are generated by using a uniform extrapolation

technique. The nonlinear viscous drag forces also contribute to the nonlinear slowly

varying motions. The time-series generation of the input wave field and the

corresponding first-order wave-frequency and second-order slowly varying wave forces

and spar motions are based on the two-term Volterra-series expansion (Kim et al., 1999;

Kim and Yue, 1991). For the design of offshore floating platforms, 3-hour simulations

are usually required for the survival condition. However, in the case of the FOWT

design, a 1-hour simulation length is usually recommended.

The natural frequencies of the Hywind spar platform are given in Table 6.4. It is

seen that all the natural frequencies are located below the lowest wave frequency of

appreciable energy except the yaw mode. However, yaw motions will be small anyway

due to the minimal wave-induced yaw moments on the vertical-cylinder hull.

77

Table 6.4 Natural frequencies of platform motions (Hywind spar)

Mode rad/s Mode rad/s

Surge 0.05 Sway 0.05

Heave 0.20 Roll 0.22

Pitch 0.22 Yaw 0.71

(a) (b) (c) Figure 6.3 Normalized mode shapes of (a) tower fore-aft, (b) tower side-to-side and

(c) blades

The flexibility of the tower is included by using a linear modal representation as

suggested in FAST. As shown in Figure 6.3, two fore-aft and two side-to-side mode

shapes of tower and two flap-wise modes and one edgewise mode of blades are used for

the coupled dynamic analysis. The natural frequencies of those elastic modes at

17.11m/s steady wind are tabulated in Table 6.5. The tower base is located at the 10m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‐5 5 15

1st mode

2nd mode

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

‐5 5 15

1st mode

2nd mode

0

0.2

0.4

0.6

0.8

1

‐1 0 1 2

Flap 1st mode

Flap 2nd mode

Edge mode

78

height from the MWL so the flexibility of the tower begins from that height. The rated

power is 5MW, and the rotor diameter is 126m.

Table 6.5 Natural frequencies of tower and blade at 17.11 m/s steady wind

Mode rad/s

1st tower fore-aft mode 2.33

2nd tower fore-aft mode 16.22

1st tower side-to-side mode 2.31

2nd tower side-to-side mode 14.34

Blade 1st flapwise 4.51

Blade 2nd flapwise 12.63

Blade 1st edgewise 6.86

6.4 Coupled Dynamic Analysis in the Time Domain

In this study, the effects of more rigorous aerodynamic loading, flexible

tower/blade, rotating blades, and blade pitch-angle control on floater motions and

mooring tensions are investigated in the time domain by comparing the coupled and

uncoupled numerical models. The coupled analysis is carried out by using the FAST-

CHARM3D hybrid program, and the tower-blade portion and floater portion are

dynamically interacting at each time step by exchanging dynamic and kinematic

information. In the uncoupled analysis, the tower-blade portion is modeled by another

rigid body with equivalent wind loading, in the way typical offshore oil and gas

platforms are analyzed. The equivalent mean wind loading on the swept area of blades is

79

determined from that of the coupled analysis by adjusting the blade drag coefficient as

shown in Table 6.6.

Table 6.6 Wind load for uncoupled dynamics

Item Value

Rotor Diameter 126 m

Swept Area 12468.98 m2

Drag Coefficient 0.168

Uncoupled Mean Wind Load 374.5 kN

Table 6.7 Environmental conditions (Hywind spar)

Item Value

Reference Wind Speed at 10m 13 m/s

Mean Wind Speed at Hub Height 17.11 m/s

Water Depth 320 m

Wave Heading 0 deg

Significant Wave Height 5.0 m

Peak Wave Period 8.69 sec

The wind and wave are collinear and their headings are fixed at 0 degree;

currents are not considered in the present study for convenience. The JONSWAP wave

spectrum is used with a significant wave height of 5 m and peak wave period of 8.69s.

As for wind, a 1-hour mean wind speed (at 10 m height) of 13 m/s is used and a time

80

dependent wind velocity is generated from the corresponding API wind spectrum. The

environmental condition is summarized in Table 6.7.

During the time-marching procedure, several control methods are working

together to maximize and optimize the power capture. In this study, blade-pitch control

and variable-speed-torque control methods are adopted. Some modifications of the

conventional control strategies typically used for land-based turbines are applied to

reduce large resonant motions and eliminate negative damping of the platform pitch

mode. Otherwise, unacceptably large resonant motions would occur because the blade-

pitch-angle-control-induced excitations act very close to pitch-heave natural frequencies.

In Figures 6.4 ~ 6.9, 6-DOFs motions of the coupled and uncoupled cases are

compared to observe the effects of rotor-tower coupling. Due to the symmetry of the hull

geometry and the head-direction of wind and wave, sway-roll-yaw motions of the

uncoupled case are zero, but the corresponding motions of the coupled case show non-

zero displacements because of the interaction between the hull and wind turbine. Due to

the aero-loading on blades and gyroscopic effects of blade rotation, there exist non-zero

mean values of sway, roll and yaw in coupled analysis.

(a) (b) Figure 6.4 Surge motion (a) and spectra (b) (Hywind spar)

0 2000 4000 6000 8000 10000-10

0

10

20

30

Time(sec)

m

Coupled

Uncoupled

0 0.5 1 1.5 20

50

100

150

200

(rad/sec)

S(

)

Coupled

Uncoupled

81

(a) (b) Figure 6.5 Sway motion (a) and spectra (b) (Hywind spar)

(a) (b) Figure 6.6 Heave motion (a) and spectra (b) (Hywind spar)

(a) (b) Figure 6.7 Roll motion (a) and spectra (b) (Hywind spar)

0 2000 4000 6000 8000 10000-1.5

-1

-0.5

0

0.5

Time(sec)

m

Coupled

Uncoupled

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

0.25

(rad/sec)

S(

)

Coupled

Uncoupled

0 2000 4000 6000 8000 10000-1.5

-1

-0.5

0

0.5

1

Time(sec)

m

Coupled

Uncoupled

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

(rad/sec)

S(

)

Coupled

Uncoupled

0 2000 4000 6000 8000 10000-0.2

0

0.2

0.4

0.6

Time(sec)

deg

Coupled

Uncoupled

0 0.5 1 1.5 20

0.005

0.01

0.015

(rad/sec)

S(

)

Coupled

Uncoupled

82

(a) (b) Figure 6.8 Pitch motion (a) and spectra (b) (Hywind spar)

(a) (b) Figure 6.9 Yaw motion (a) and spectra (b) (Hywind spar)

An interesting phenomenon we can observe is that the uncoupled surge and pitch

responses are slightly greater than the corresponding coupled responses. For instance,

the maximum surge displacement in the uncoupled case is 21.7m, but that of the coupled

case is only 20.3m. A similar trend can be observed in pitch responses. The standard

deviations of surge and pitch motions in the uncoupled case are 48% and 14% higher

than those of the coupled case, respectively. This trend is actually opposite to that of a

TLP-type FOWT (Bae and Kim, 2011). This phenomenon can be explained by the

blade-pitch-control action of the coupled case. Assuming that equivalent winds are

applied to the rotor for both cases, the platform of the uncoupled case will be fully

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10

Time(sec)

deg

Coupled

Uncoupled

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5

10

15

(rad/sec)

S(

)

Coupled

Uncoupled

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0

0.5

1

Time(sec)

deg

Coupled

Uncoupled

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0.01

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0.04

0.05

(rad/sec)

S(

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Coupled

Uncoupled

83

actuated by the incident wind. However, in the coupled case, the turbine starts to adjust

its blade-pitch angle to regulate the incoming wind effect. This pitch-to-feather action

plays a role in mitigating the wind loading and the corresponding platform response.

Moreover, in the rotor-floater coupled analysis, the relative wind velocity with respect to

the platform motion is used to calculate the wind loading but the relative-wind-velocity

effect is ignored in the uncoupled analysis. Also, in the coupled analysis, the

instantaneous wind loading is applied at the instantaneous tower-blade position thus

acting in all directions including heave direction. On the other hand, in the uncoupled

analysis, the wind loading is applied only to the mean position of tower and blade, and

thus only the horizontal wind loading (and the corresponding pitch moment) is applied to

the center of the equivalent disk. For this reason, the coupled heave motions are

appreciably greater than the uncoupled heave motions as can be seen in Figure 6.6. In

the coupled case, there exist non-zero transverse motions due to the gyroscopic effect of

blade rotation and the influence of the blade pitch-angle control. This phenomenon

cannot be obtained from the uncoupled analysis. The statistics of hull responses are

tabulated in Table 6.8.

Table 6.8 Platform motion statistics (Hywind spar)

Max. Min Mean SD

Surge (m)

Uncoupled 2.17E+01 4.14E+00 1.20E+01 3.04E+00

Coupled 2.03E+01 6.77E+00 1.23E+01 2.06E+00

Sway (m)

Uncoupled 8.61E-06 -9.53E-06 -7.30E-08 2.29E-06

Coupled -3.32E-01 -1.05E+00 -6.27E-01 1.21E-01

84

Table 6.8 Continued

Max. Min Mean SD

Heave (m)

Uncoupled 1.97E-01 -3.58E-01 -7.99E-02 7.54E-02

Coupled 4.58E-01 -8.33E-01 -2.33E-01 1.68E-01

Roll (deg)

Uncoupled 3.52E-06 -3.16E-06 -1.01E-08 9.86E-07

Coupled 4.21E-01 1.77E-01 2.96E-01 3.72E-02

Pitch (deg)

Uncoupled 6.72E+00 -5.16E-01 2.63E+00 8.81E-01

Coupled 6.35E+00 -1.12E-01 2.76E+00 7.75E-01

Yaw (deg)

Uncoupled 9.17E-07 -8.87E-07 -2.13E-08 1.12E-07

Coupled 3.39E-01 -4.58E-01 -8.74E-02 1.07E-01

The differences in hull motions between the coupled and uncoupled cases

directly affect the top-tension statistics of tethers which are summarized in Table 6.9.

The mooring lines arrangement is depicted in Figure 6.10. The upwind-side lines such as

lines #2 and #3 will have higher tensions as can be seen in Figures 6.12 ~ 6.13, and the

tension of downwind-side line #1 in Figure 6.11 will be decreased due to the surge offset.

Due to more severe surge slow-drift motions in the uncoupled analysis, the maximum

top tensions of lines are increased by 3~6%.

85

Figure 6.10 Top view of mooring-line arrangement (Hywind spar)

(a) (b) Figure 6.11 Top-tension (a) and spectra (b) of Line #1 (Hywind spar)


0 2000 4000 6000 8000 10000500

600

700

800

900

1000

Time(sec)

kN

Coupled

Uncoupled

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2

4

6

8x 10

4

(rad/sec)

S(

)

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1000

1100

1200

1300

1400

Time(sec)

kN

Coupled

Uncoupled

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

WIND WAVE

Line #1

Line #2

Line #3

86


Table 6.9 Mooring line top tension statistics (Hywind spar)

Max. Min Mean SD

Line #1 (kN)

Uncoupled 9.52E+02 6.07E+02 7.65E+02 5.34E+01

Coupled 8.98E+02 6.41E+02 7.59E+02 3.73E+01

Line #2 (kN)

Uncoupled 1.33E+03 9.51E+02 1.13E+03 5.62E+01

Coupled 1.29E+03 9.94E+02 1.13E+03 4.13E+01

Line #3 (kN)

Uncoupled 1.33E+03 9.51E+02 1.13E+03 5.62E+01

Coupled 1.27E+03 9.89E+02 1.12E+03 4.21E+01

Fore-aft accelerations at 3-different locations of the tower were also investigated

and the coupled and uncoupled cases are compared in Figures 6.14 ~ 6.16. For the

coupled case, the total acceleration at a given height is calculated by the summation of

the local tower acceleration from elastic vibration and the global acceleration due to the

hull motion. Phase differences between the local-tower acceleration and global

acceleration were considered and included in the calculation of the total acceleration. In

the uncoupled analysis, the entire system is treated as a rigid body so only the global

accelerations are considered at the respective heights.

0 2000 4000 6000 8000 10000900

1000

1100

1200

1300

1400

Time(sec)

kN

Coupled

Uncoupled

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2

4

6x 10

4

(rad/sec)

S(

)

87

(a) (b) Figure 6.14 Tower-acceleration (a) and spectra (b) at 85.66m from MWL



The statistics of the tower fore-aft accelerations in Table 6.10 show that the

maximum acceleration of the coupled analysis is increased by 10.4% at the top position

0 2000 4000 6000 8000 10000-2

-1

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1

2

Time(sec)

m/s

2

Coupled

Uncoupled

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0.2

0.4

0.6

0.8

(rad/sec)

S(

)

Coupled

Uncoupled

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-1

0

1

2

Time(sec)

m/s

2

Coupled

Uncoupled

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0.2

0.4

0.6

0.8

(rad/sec)

S(

)

Coupled

Uncoupled

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-1

0

1

2

Time(sec)

m/s

2

Coupled

Uncoupled

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0.4

0.6

0.8

(rad/sec)

S(

)

Coupled

Uncoupled

88

compared to the uncoupled case. On the contrary, the maximum uncoupled acceleration

of tower at the middle position is increased by 23.8% compared to the coupled case.

Similarly, the maximum uncoupled tower acceleration is larger than that of the coupled

case by 8.9% at the near bottom position. These phenomena can be explained by the

tower elastic modes of the coupled analysis. The tower top is accelerated more by tower

elastic bending modes, while the tower base is accelerated mostly by the surge

acceleration of the platform itself.

Table 6.10 Tower acceleration statistics (Hywind spar)

Max. Min Mean SD

85.66m (m/s2)

Uncoupled 1.63E+00 -1.76E+00 1.06E-04 4.48E-01

Coupled 1.80E+00 -1.89E+00 9.46E-05 4.56E-01

58.50m (m/s2)

Uncoupled 1.39E+00 -1.51E+00 8.71E-05 3.83E-01

Coupled 1.14E+00 -1.22E+00 1.83E-04 2.97E-01

11.94m (m/s2)

Uncoupled 9.76E-01 -1.07E+00 5.48E-05 2.72E-01

Coupled 9.28E-01 -9.82E-01 8.60E-05 2.42E-01

6.4 Influence of Control Strategies

6.4.1 Two Control Strategies

For the NREL 5MW turbine, two control systems are designed to work. A

generator-torque controller and a blade-pitch controller are working in the below-rated

and above-rated wind-speed range respectively.

89

Figure 6.17 Two control strategies (Hywind spar)

The generator-torque controller is designed to maximize power capture and the

blade-pitch controller is designed to regulate generator speed by gain-scheduled

proportional-integral (PI) control. The schematic diagram of the two control strategies is

depicted in Figure 6.17.

The controllers determine its feedback order such as generator torque of blade-

pitch angle by measuring the filtered shaft speed. The measured shaft speed is then

compared with the target shaft speed. The error between measured and target shaft speed

can be expressed as the equation of motion for the rotor-speed error.

02

0 0 0

0

1 1

10

Drivetrain Gear d Gear p

Gear i

PP PI N K N K

PN K

90

where DrivetrainI is a drivetrain inertia and P and 0P are mechanical power and rated

mechanical power respectively. is a full-span rotor-collective blade-pitch angle and

0 is a rated low-speed shaft rotational speed. /P stands for a sensitivity of

aerodynamic power to rotor-collective blade pitch. dK , pK and iK are the blade-pitch

controller proportional, integral, and derivative gains respectively.

It is known that the rotor-speed-error responds as a second-order system as

shown in the above equation (Jonkman, 2008). For the 5MW baseline, NREL

recommended the optimal gain values of proportional ( pK = 0.01882681s) and integral

( iK = 0.008068634) gains at a minimum blade pitch setting for the baseline wind

turbines. The derivative gain dK is set to zero because it gives better performance than

other values. Based on these gains, the blade-pitch control system uses a new gain

according to the blade pitch angle input. Note that the negative damping represented by

the 20 0/P term is introduced in the speed error response and should be compensated

by the proportional gain in the blade-pitch controller.

6.4.2 Modification of Control Strategies

The blade-pitch response of this control strategy can be evaluated for a land-

based turbine. The step variation of input wind speed is applied to the land-based turbine

and the response of blade-pitch angle is investigated. The current control parameter

(conventional control strategy) gives a very fast and accurate response of blade-pitch

angle. This strategy is very good for land-based turbines or TLP-type offshore wind

91

turbines since they have minimal rotational motion and very high pitch natural

frequencies (Bae and Kim, 2011). However, if the pitch/roll natural frequency of a

floating platform is low and close to the pitch-angle-actuator frequency, such as spar-

type or semisubmersible-type FOWTs, the interaction between the platform pitch motion

and the variation of thrust force due to the blade-pitch control action may cause serious

resonance. In order to avoid this resonance, the pitch-angle-actuator frequency must be

lowered by detuning the gain values.

(a) (b) Figure 6.18 Step wind input (a) and blade pitch angle (b) (Land-based)

According to Larsen and Hanson (Larsen and Hanson, 2007), the lowest

controller-response natural frequency should be less than the platform’s pitch natural

frequency to avoid the negatively damped platform motion. By detuning gain values,

one can reduce the controller-response natural frequency and ensure the platform

motions remain positively damped. In Figure 6.18(b), the pitch angle of conventional

control reaches its target pitch angle very fast in response to the variation of step input

wind speed. If we reduce the gain values, then the overall reaction of the pitch-angle

actuator is changed. It takes longer to reach the target pitch angle and the gradient of

0 50 100 150 200 250 30010

15

20

25

Time(sec)

m/s

0 50 100 150 200 250 3000

5

10

15

20

Time(sec)

deg

Conventional Control

Detuned gain Control

92

pitch angle variation is small compared to that of conventional control. When it comes to

control quality, the conventional control shows a larger transient overshoot in the lower

wind-speed range, while the modified gain control shows relatively smaller or no

transient overshoots.

In addition to this detuned-gain modification, the negative damping term can also

be reduced to zero if the variable-speed-torque control changes the Region 3 from a

constant generator power to a constant generator-torque. Region 3 is one of the control

regions where the generator torque is computed as a tabulated function of the filtered

generator speed; it was originally designed to produce a constant generator power. This

modification may reduce the negative damping term of the speed-error equation but it

could also affect the quality of the generated power output. With the same step variation

of input wind speed, and keeping the same gain values, the trends of conventional and

constant-torque control are compared.

(a) (b) Figure 6.19 Generator torque (a) and generator power (b) (Land-based) Figure 6.19(a) shows that the modification of Region 3 produces constant

generator torque regardless of the change of input wind speed. However, the generated

0 50 100 150 200 250 3000

10

20

30

40

50

Time(sec)

kN.m


Constant torque Control

0 50 100 150 200 250 3000

2000

4000

6000

Time(sec)

kW


Constant torque Control

93

power has a relatively larger overshoot at every initial stage of wind-speed variation

compared to the conventional-control case. This kind of power surge may have negative

effects on the generator or other electric devices in the turbine.

As already pointed out, the two modifications of control strategy explained

earlier are in fact not necessary for the land-based or TLP-type wind turbines. However,

in the case of the Hywind spar, the modifications are quite essential since the hull

motions can be greatly amplified without them. This is particularly so since the

conventional-control-induced excitation frequencies are very close to the surge/sway and

roll-pitch natural frequencies. However, by modifying the control strategy as explained

above, the detrimental resonance effects can be avoided. In order to see the effects of

modifications on the Hywind spar, similar tests are carried out with the same step wind

input.

(a) (b) Figure 6.20 Step wind input (a) and blade pitch angle (b) (Hywind spar) Figure 6.20(b) shows the comparison of the pitch-angle variation between the

conventional and modified control strategies. We can see that the pitch angle changes

very hastily with significantly amplified amplitudes with the conventional-control

0 50 100 150 200 250 30010

15

20

25

Time(sec)

m/s

0 50 100 150 200 250 3000

5

10

15

20

25

Time(sec)

deg


Modified Control

94

strategy. The conventional-control-induced excitation frequencies are very close to the

pitch natural frequency of the Hywind spar. As a result, large platform pitch responses

occur, as can be seen in Figure 6.21. The blade-pitch controller tries to catch up with the

variation of input wind and it consequently produces a variation of thrust force in the

frequency range similar to platform pitch resonance.

Figure 6.21 Platform pitch motion for two control strategies

(a) (b) Figure 6.22 Generator torque (a) and generator power (b) for two control strategies

To avoid this kind of resonance, the modified control strategy is applied to the

same Hywind spar platform; it is seen in Figure 6.21 that the platform pitch response is

greatly reduced as a result. However, the generator power surge of the modified control

0 50 100 150 200 250 300-5

0

5

10

15

Time(sec)

deg


Modified Control

0 50 100 150 200 250 3000

10

20

30

40

50

Time(sec)

kN.m


Modified Control

0 50 100 150 200 250 3000

2000

4000

6000

Time(sec)

kW


Modified Control

95

may have a negative effect (see Figure 6.22(b)) as a result as explained above. Thus a

further check needs to be done to deal with this effect.

6.4.3 Response in Random Sea Environment

So far, the effect of control strategy on the global performance of FOWTs has

been investigated under the simple step-wind environment. In this section, we have

considered a typical wind-wave environment as a more realistic random input to the

respective control systems. The time-varying wind speed at hub height is generated

based on the API wind spectrum ranging 5s to 3,600s. If the aerodynamic loading is to

be generated in a strict manner, the full-wind-field data inside the blade-swept area need

to be used, but in the present study only the variation of the wind velocity in the vertical

direction is considered assuming that the sideways variation can be neglected. The

random waves are generated from the JONSWAP spectra and 1-hour simulations were

carried out. Total time domain simulation is for 4,000s, including initial 400s of ramp

time. Statistics are obtained based on those time series from 400s to 4,000s after

eliminating the effect of initial transient responses. The environmental condition used for

the simulation is the same as the previous section in Table 6.7.

In Figure 6.23, the blue (solid) line shows the variation of blade pitch angle as a

result of applying the conventional control strategy. It shows a lot of fluctuation. From

time to time, the blade pitch angle hits 0 degree which means the controller tries to

capture a maximum lift force from the blade. This random pitch-angle action is primarily

concentrated in the frequency range between 0.15~0.23 rad/s. The frequency range

96

coincides with the pitch/roll/heave natural frequencies of the Hywind spar (see Table

6.4). Therfore, it is expected that large pitch/heave motions will occur as a result of the

blade-pitch-angle control.

(a) (b) Figure 6.23 Blade pitch angle (a) and spectra (b) for two control strategies

The corresponding time history of the thrust force measured at the low-speed

shaft point is given by the blue line in Figure 6.24. At this point, the thrust force is

affected by the aerodynamic force which is regulated by the blade-pitch controller. The

trend is very similar to the blade pitch-angle variation. The high peak is also shown at

the same frequency range.

If the two modifications are applied to the retuned control system, the duty cycle

of blade-pitch is reduced noticeably, as can be seen on the dotted red line in Figure 6.23.

The actuation frequency is much lower than the pitch-roll-heave resonance frequencies.

The resulting pitch motions will be smaller, so the blade-pitch controller needs to spend

less effort to adjust. Compared to the conventional control case in Figure 6.23(b), the

peak frequency is located at a much smaller frequency.

0 1000 2000 3000 40000

5

10

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20

25

Time(sec)

deg


Modified Control

0 0.5 1 1.5 20

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200

300

400

500

(rad/sec)

S(

)


Modified Control

97

(a) (b) Figure 6.24 Shaft thrust force (a) and spectra (b) for two control strategies

The major contribution of the thrust force is aerodynamic loading. The thrust

force directly affects the floater pitch motion. In the case of conventional control, the

blade-action-induced thrust force is again greatly amplified in the range of 0.15~0.23

rad/s for the same reason. The harmful resonance disappears when the modified control

scheme is applied as shown in Figure 6.24.

The same kind of improvement of performance can also be seen in the 6-DOFs

platform motions (Figures 6.25 ~ 6.30) by applying the modified control strategy.

Without such modification, the platform motions become too large, especially in heave

and pitch modes, so they are not acceptable in the design. The sway, roll and yaw are

also appreciably influenced by the blade-control action. The results typically illustrate

that the blade-control scheme strongly influences platform motions and the phenomenon

can only be explained by use of the rotor-floater-mooring fully-coupled time-domain

simulation program. The same phenomenon has also been observed in an experiment

with spar-type FOWT (Nielsen et al., 2006).

0 1000 2000 3000 4000-500

0

500

1000

1500

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

5

10x 10

5

(rad/sec)

S(

)


Modified Control

98

(a) (b) Figure 6.25 Surge motion (a) and spectra (b) for two control strategies

(a) (b) Figure 6.26 Sway motion (a) and spectra (b) for two control strategies

(a) (b) Figure 6.27 Heave motion (a) and spectra (b) for two control strategies

0 1000 2000 3000 4000-10

0

10

20

30

Time(sec)

m


Modified Control

0 0.5 1 1.5 20

200

400

600

(rad/sec)

S(

)


Modified Control

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-2

-1

0

1

Time(sec)

m


Modified Control

0 0.5 1 1.5 20

1

2

3

4

(rad/sec)

S(

)


Modified Control

0 1000 2000 3000 4000-5

0

5

Time(sec)

m


Modified Control

0 0.5 1 1.5 20

50

100

150

(rad/sec)

S(

)


Modified Control

99

(a) (b) Figure 6.28 Roll motion (a) and spectra (b) for two control strategies

(a) (b) Figure 6.29 Pitch motion (a) and spectra (b) for two control strategies

(a) (b) Figure 6.30 Yaw motion (a) and spectra (b) for two control strategies A gyroscopic effect can also be seen in the case of conventional control in Figure

6.30. This gyroscopic yaw moment comes not from the aerodynamic loads on the rotor

but from the spinning inertia of the rotor combined with large pitch motion.

0 1000 2000 3000 4000-1

0

1

2

Time(sec)

deg


Modified Control

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

(rad/sec)

S(

)


Modified Control

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-5

0

5

10

15

Time(sec)

deg


Modified Control

0 0.5 1 1.5 20

100

200

300

400

(rad/sec)

S(

)


Modified Control

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-2

0

2

Time(sec)

deg


Modified Control

0 0.5 1 1.5 20

2

4

6

8

(rad/sec)

S(

)


Modified Control

100

Table 6.11 shows the statistics of the Hywind spar motion in the given random

environment. All the 6-DOFs motions with conventional control strategy show very

large maximum and standard-deviation values due to the control-actuated pitch/heave

resonance.

Table 6.11 Statistics of platform motion in two control strategies

Max. Min Mean SD

Surge (m)

Conventional 2.76E+01 2.21E+00 1.39E+01 6.76E+00

Modified 1.87E+01 6.52E+00 1.22E+01 1.98E+00

Sway (m)

Conventional 5.38E-01 -2.09E+00 -7.03E-01 5.44E-01

Modified -3.64E-01 -1.02E+00 -6.34E-01 1.17E-01

Heave (m)

Conventional 4.83E+00 -4.74E+00 -5.04E-01 2.84E+00

Modified 1.71E-01 -9.21E-01 -3.64E-01 1.75E-01

Roll (deg)

Conventional 1.52E+00 -6.08E-01 3.84E-01 4.36E-01

Modified 4.03E-01 1.97E-01 2.97E-01 3.44E-02

Pitch (deg)

Conventional 1.30E+01 -5.79E+00 3.26E+00 5.55E+00

Modified 5.05E+00 6.78E-01 2.74E+00 7.10E-01

Yaw (deg)

Conventional 1.56E+00 -3.08E+00 1.77E-02 9.48E-01

Modified 3.41E-01 -4.95E-01 -9.38E-02 1.11E-01

The comparisons of the rotor speed and generated power output between the two

cases are also shown in Figures 6.31 ~ 6.32. The rotor speed with conventional control is

also greatly affected by platform pitch resonance combined with blade pitch actuation.

The power output from the conventional control had numerous power drops during the

simulation time. The reason for this sudden drop is the instantaneous reduction in the

101

relative wind speed due to large pitch backward motions. Thus, the time frame of these

sudden drops coincides with that of the 0-degree blade-pitch angle. In the case of the

modified control, the number and range of power drops are significantly reduced but a

nontrivial power overshoot also exists as a minor side effect. Nevertheless, the overall

quality of the generated power with the modified control strategy is much better than

that of the other case.

(a) (b) Figure 6.31 Rotor speed (a) and spectra (b) for two control strategies

(a) (b) Figure 6.32 Generator power (a) and spectra (b) for two control strategies

The fore-aft shear force, axial force, and fore-aft bending moment at the tower

base (Figures 6.33 ~ 6.35) are important to the structural design of a tower. The general

0 1000 2000 3000 4000-5

0

5

10

15

Time(sec)

rpm


Modified Control

0 0.5 1 1.5 20

5

10

15

(rad/sec)

S(

)


Modified Control

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2000

4000

6000

Time(sec)

kW


Modified Control

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4

6

8x 10

5

(rad/sec)

S(

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Modified Control

102

tendency of the time histories of the shear, axial forces, and bending moment at the

tower base is similar to that of platform motion. It is seen that large forces and moments

are transferred to the position in case of the conventional control and the location of the

peak is consistent with that of platform motion. The maximum shear force and bending

moment with the conventional control is more than 70% higher than that of the modified

control. The higher standard deviation of the shear force means more vulnerability to

fatigue failure which may happen when the blade control system is poorly designed. The

negative sign of axial force stands for the compression force that may be a concern for

buckling failure.

(a) (b) Figure 6.33 Tower-base fore-aft shear force (a) and spectra (b) for two control

strategies

(a) (b) Figure 6.34 Tower-base axial force (a) and spectra (b) for two control strategies

0 1000 2000 3000 4000-2000

0

2000

4000

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

5

10

15x 10

6

(rad/sec)

S(

)


Modified Control

0 1000 2000 3000 4000-6100

-6000

-5900

-5800

-5700

-5600

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

5

10

15x 10

4

(rad/sec)

S(

)


Modified Control

103

(a) (b) Figure 6.35 Tower-base pitch bending moment (a) and spectra (b) for two control

strategies

The structural loading on the blade root location was also investigated in this

study. Since the configuration of the blades attached to the rotor hub is a kind of

cantilever beam, the highest shear force and bending moment are expected at the blade-

root location. Based on the elastic blade configurations, two shear forces, flapwise and

edgewise, at the root location were selected for comparison as shown in Figures 6.36 ~

6.37.

(a) (b) Figure 6.36 Flapwise shear force at blade root (a) and spectra (b) for two control

strategies

0 1000 2000 3000 4000-2

0

2

4x 10

5

Time(sec)

kN.m


Modified Control

0 0.5 1 1.5 20

2

4

6x 10

10

(rad/sec)

S(

)


Modified Control

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0

200

400

600

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)


Modified Control

104

In the frequency domain, the flapwise shear force with conventional control

shows a high peak around the platform pitch resonance frequency, and small peaks

around 1P frequency of 1.27 rad/s. 1P represents the once per revolution frequency of

the rotor. In case of modified control, 1P frequency is dominant.

(a) (b) Figure 6.37 Edgewise shear force at blade root (a) and spectra (b) for two control

strategies In the case of edgewise direction in Figure 6.37, the shear force is more strongly

associated with the rotation of the blade. This shear force shows a clear peak at the 1P

frequency. The shear forces in the frequency domain show a different trend between the

two control strategies. The modified control shows a smoother transition around the 1P

frequency while the conventional control shows sharper and higher peaks at 1P

frequency with minor peaks nearby. These differences are mostly due to the different

actuator speed of the blade pitch which results in a smooth transition with a low speed

actuator (modified) and a sharper transition with a rapid actuator (conventional). The

maximum flapwise shear force with the conventional control is nearly 39% greater than

that of the modified control.

0 1000 2000 3000 4000-300

-200

-100

0

100

200

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

5

10

15x 10

4

(rad/sec)S

( )


Modified Control

105

The differences in hull motions between the conventional control and modified

control directly affect the top-tension statistics of mooring lines. The mooring-line

arrangement is depicted in Figure 6.10. The wind-wave direction is along the x (in the

direction of line #1 and between taut-side lines #2 and #3). The upwind-side lines such

as line #2 and #3 will have higher tensions and the tension of the downwind-side line #1

will be smaller due to the mean surge offset

(a) (b) Figure 6.38 Top-tension of Line #1 (a) and spectra (b) for two control strategies

(a) (b) Figure 6.39 Top-tension of Line #2 (a) and spectra (b) for two control strategies

The standard deviations of top tensions from upwind lines with modified control

are 18 ~ 21% greater than those of the conventional control case. This trend is quite

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600

700

800

900

1000

Time(sec)

kN


Modified Control

0 0.5 1 1.5 20

5000

10000

15000

(rad/sec)

S(

)


Modified Control

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1000

1100

1200

1300

1400

Time(sec)

kN


Modified Control

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5000

10000

15000

(rad/sec)

S(

)


Modified Control

106

opposite to the previous results. An increase at the lowest frequency due to modified

control can be expected since the blade-action was intentionally moved to lower

frequency. The three lowest natural frequencies of the mooring lines are estimated to be

0.36, 0.72, and 1.07 rad/s. In this case, the peak around 1.07 rad/s in Figures 6.38 ~ 6.39

corresponds to the third lowest mooring dynamics mode. Interestingly, the peak of

modified control is slightly larger than that of the conventional control. In terms of

mooring line tension, the modified control is less efficient than the conventional control

despite the significant advantage in floater motions. These responses can be captured

only by a full mooring dynamics model; an alternative quasi-static mooring analysis

model cannot get those high peaks as can be seen in Figure 6.40.

(a) (b) Figure 6.40 Spectra of top-tension of Line #1 (a) and Line #2 (b) for quasi-static

and FE mooring 6.5 Discussion

In case of the Hywind spar, it is seen that the rotor-floater coupling effects

decrease the maximum horizontal offsets and mooring-line tensions but heave motions

are increased due to the time-varying wind loading caused by the time-varying blade

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Quasi-Static Mooring

FE Mooring

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Quasi-Static Mooring

FE Mooring

107

pitch angle and relative wind velocity against the moving tower. The tower top

accelerations are, however, increased due to the effects of tower flexibility which in turn

will greatly affect the corresponding inertial loading on nacelle and blades. This may be

of important concern for the structural robustness and fatigue life of the system.

In addition, the influence of two different control strategies (conventional and

modified) on the global performance of floating offshore wind turbines, especially for a

Hywind spar platform, was investigated. The conventional control scheme that gives fast

and accurate feedback response is designed for land-based or TLP-type wind turbines. If

the pitch stiffness of a FOWT is small, such as that of spar-type or semi-submersible-

type floaters, the control-induced excitations may cause resonance which can

significantly increase the floater responses. This was clearly demonstrated in the present

time-domain simulations by using a fully coupled dynamic analysis program. In such a

case, a modified control strategy is necessary to avoid the resonance effect. By detuning

the gain values and modifying the Region 3 control method, better control strategy can

be devised. Applying the modified control strategy showed that the 6-DOFs floater

motions and tower-base/blade-root shear forces and bending moments are noticeably

reduced and the corresponding turbine performance, such as generated power quality, is

also appreciably improved. However, the modified control may induce slightly higher

mooring line tension. In conclusion, the time-domain aero-elastic-control-floater-

mooring coupled dynamic analysis computer program was successfully developed. It can

be used for checking the global performance and robustness of any type of FOWTs for

108

any kind of environment, control scheme, and mooring system and can also be very

helpful in the design of more reliable and innovative FOWTs in the future.

109

7. CASE STUDY II: THREE-TURBINE SEMI-SEBMERSIBLE

7.1 Introduction

In the previous chapters, the theory and numerical methods for multiple turbines

on one platform were discussed. By expanding the coefficient matrix rsC and the

number of degrees of freedom, the FAST-CHARM3D coupled tool can analyze the

multiple turbines on one platform, including tower and blade elasticity in time domain.

Since there is no real MUFOWT structure at this moment, the floating platform and

mounted turbines were selected considering their hydrostatic stability and turbine size.

As a preprocessor, WAMIT was used to obtain hydrodynamic coefficients such as added

mass, radiation damping and first-order wave force.

In terms of aero dynamics, the effects of dynamic aero loadings were considered

separately in individual turbines. However, the aerodynamic interference and shade

effect between adjacent turbines were not considered in this study because of the

complexity and uncertainty of their aerodynamic effect on the turbulent wind field.

Instead, the turbines were positioned with enough spacing in side-by-side and fore-aft

directions. The water depth was set to 300m, and 6 catenary mooring lines were attached

in the fairlead position to avoid platform drift in a wind-wave-current environment. In

this case study, two 1.5MW turbines and one 5MW turbine were adopted and arranged

on top of a triangular-shaped semi-submersible platform. Overall platform responses

including mooring line tension and turbine performances were assessed. To see the

integrated platform responses, one turbine fault scenario was simulated and the resultant

110

overall platform responses as well as the performances of other turbines were checked

and presented.

7.2 Configuration of the Platform, Mooring System and Turbines

The floating platform used in this case study was triangular with three vertical

columns and three horizontal pontoons as seen in Figure 7.1. The left two columns

support 1.5MW turbines each, and the right column supports one 5MW turbine. The

properties of 5MW turbine are in Table 6.1, and the properties of 1.5MW turbine are

specified in WindPACT studies conducted by NREL (Malcolm et al., 2002).

To satisfy the static equilibrium of the platform in calm water, the ballast water

was filled inside the platform.

(a) (b)

Figure 7.1 Triangular platform geometry (a) and system configuration (b) Since the weight of the two front 1.5MW turbines was lighter than that of rear

5MW turbine, more ballast water should be filled out in the front columns. The column

diameter was 10m each, and the distance between each column was 79.67m. The

111

platform draft was 20m, and the width and height of each pontoon was 6m and 5m

respectively. Figure 7.2 shows the dimensions of the triangular platform.

(a) (b) Figure 7.2 Platform dimensions from top (a) and side (b) (Triangular platform)

The platform hull weight was estimated with the submerged surface area and

given plate thickness (15mm). The mass properties of the platform, including mass

moment of inertia, were calculated considering the displaced steel platform and ballast

water. The details of the platform properties are tabulated in Table 7.1.

Table 7.1 Specification of triangular platform

Item Unit Value


Column Diameter m 10.0

Length between Columns m 79.67

Pontoon Width m 6.0

Pontoon Height m 5.0

Platform Weight, including Ballast N 98,246,039

112

Table 7.1 Continued

Item Unit Value

Platform Buoyancy N 113,607,091

Platform CM Location Below SWL m 15.0

Platform Roll Inertia about CM kg·m2 7,051,945,158


Platform Yaw Inertia about Platform Centerline kg·m2 13,099,535,514

In case of the mooring system, two catenary mooring lines were installed at each

corner, making a total of six mooring lines in this system. Each mooring line consisted

of a chain-steel wire-chain combination with a total length of 644.6m. The mooring line

top tensions were estimated based on the equilibrium relation between the platform

weight and buoyancy. The details of the mooring system properties are tabulated in

Table 7.2.

Table 7.2 Specification of mooring system (Triangular platform)

Item Unit Value

Number of Mooring Lines ea 6

Angle Between Adjacent Lines deg 60

Depth to Anchors Below SWL (Water Depth) m 300

Depth to Fairleads Below SWL m 20.0

Unstretched Mooring Line Length m 644.6

Mooring Line Diameter (Chain) m 0.382

Mooring Line Mass Density (Chain) kg/m 381.374

Mooring Line Mass in Water (Chain) kg/m 322.638

113

Table 7.2 Continued

Item Unit Value

Mooring Line Extensional Stiffness (Chain) kN 1.328E6

Mooring Line Diameter (Steel wire) m 0.163

Mooring Line Mass Density (Steel wire) kg/m 90.041

Mooring Line Mass in Water (Steel wire) kg/m 79.334

Mooring Line Extensional Stiffness (Steel wire) kN 240192

The arrangements of six mooring lines are presented in Figure 7.3.

Figure 7.3 Mooring line configurations (Triangular platform) 7.3 Hydrodynamic Coefficients in the Frequency Domain

The hydrodynamic coefficients, including added mass, radiation damping, and

linear wave forces, were obtained by WAMIT. Figure 7.4 shows the discretized panel

distribution of the floater. The submerged body has one plane of symmetry and each side

has 1,155 panels. Second-order sum and difference frequency effects were not included

in this simulation for simplicity. The viscous drag force of the hull and mooring line was

114

not considered in the frequency domain analysis, but will be accounted for in the

following time domain analysis using Morison’s equation.

Figure 7.4 Discretized panel model of floating body (Triangular platform)

The hydrodynamic added mass and radiation damping in frequency domain are

presented in Figures 7.5 and 7.6 respectively.

Surge Roll

Figure 7.5 Added mass (Triangular platform)

-20

0

20

40

-40

-20

0

20

40

-20

-10

0

xy

z

0.5 1 1.5 24

6

8

10x 10

6

(rad/sec)

Add

ed M

ass

A11

0.5 1 1.5 26.5

7

7.5

8x 10

9

(rad/sec)

Add

ed M

ass

A44

115

Sway Pitch

Heave Yaw

Figure 7.5 Continued

Surge Roll

Figure 7.6 Radiation damping (Triangular platform)

0.5 1 1.5 24

6

8

10x 10

6

(rad/sec)

Add

ed M

ass

A22

0.5 1 1.5 26.5

7

7.5

8x 10

9

(rad/sec)

Add

ed M

ass

A55

0.5 1 1.5 21.1

1.15

1.2

1.25

1.3x 10

7

(rad/sec)

Add

ed M

ass

A33

0.5 1 1.5 20.5

1

1.5

2x 10

10

(rad/sec)A

dded

Mas

s A

66

0.5 1 1.5 20

1

2

3x 10

6

(rad/sec)

Dam

ping

B11

0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

Dam

ping

B44

116

Sway Pitch

Heave Yaw

Figure 7.6 Continued

7.4 Time Marching Simulation in Normal Operational Condition

So far, the mass and stiffness properties of the floating platform and wind turbine

system were determined and the hydrodynamic coefficients were also obtained from

WAMIT. To see the dynamic responses of MUFOWT, floater-rotor-mooring coupled

dynamic analysis of multiple turbines on one platform was carried out in the time

domain. By performing this time domain analysis, a more rigorous analysis was made of

the entire wind turbine system including blade aero dynamics with time-varying random

wind field, elastic modes of tower and blades, FEM based mooring lines and nonlinear

viscous loading of platform.

0.5 1 1.5 20

1

2

3x 10

6

(rad/sec)

Dam

ping

B22

0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

Dam

ping

B55

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(rad/sec)

Dam

ping

B33

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(rad/sec)D

ampi

ng B

66

117

7.4.1 Viscous Damping Modeling

One of the advantages of the time domain analysis is that a numerical model can

include the viscous damping effect. In FAST-CHARM3D, the viscous damping is

calculated by placing an equivalent truss or plate members inside the submerged

platform model where the diameter of the member is small compared to the wave length.

To include this effect, the Morison’s formula in Equation (7.1) was used.

1( )

2n m n a n d n n n nF C u C x SC u x u x (7.1)

The symbols and S represent the displaced volume and projected area; the fluid

density is . aC is the added mass coefficient, mC is the inertia coefficient and dC is

the drag coefficient. nu and nu are the acceleration and velocity of fluid normal to the

body, and nx and nx are the acceleration and velocity of a floating body in the normal

direction, respectively.

In the case of the three-turbine platform, the viscous drag force of the hull was

represented by employing three truss members for each column; this acted along every

normal direction of the column, and two plate members (vertical and horizontal) for each

pontoon, which act in the normal direction of the plate as well. The drag coefficient dC

was taken to be 0.6 for the cylindrical column and 1.28 for the rectangular pontoon. The

first two inertial loading terms in Equation (7.1) were not used because the incident-

diffraction potential force and hull added mass were already calculated and included in

Equation (4.54).

118

In addition to the hull viscous members above, the dynamic loadings on tethers

were also calculated by Morison’s equation. For the calculation of the loading on tethers,

the inertia coefficient mC in Equation (7.1) was taken to be 2 and the drag coefficients

dC were set to 1.3 for the wire and 2.4 for the chain. The drag coefficients for all the

Morison members are tabulated in Table 7.3.

Table 7.3 Drag coefficients of Morison members (Triangular platform)

Location dC Viscous model No. of members

Columns 0.6 Truss 3

Pontoons 1.28 Horizontal plate 3

Vertical plate 3

Chain Mooring 2.4 Slender rod 6

Wire Mooring 1.3 Slender rod 6

7.4.2 System Identification and Free Decay Test

To check the 6-DOFs platform natural frequencies, free decay tests in the time

domain were carried out. In the time domain analysis, nonlinear viscous drag by

platform and mooring line dynamics were included, but the tower and blades elasticity

were not considered and every part of turbine was assumed to be a rigid body without

any flexibility. In addition to that, the environmental loadings were not applied, which

means the free decay test was simulated in calm water without wind and waves. Figure

7.7 shows the time history and spectra of the free decay test in every mode, respectively.

119

Surge Roll

Sway Pitch

Heave Yaw

Figure 7.7 Free decay test (Triangular platform) For surge, sway and heave tests, the platform was initially moved to a 10m

position along the positive direction then released at that position. In the case of roll,

pitch and yaw, the platform was released from the positive 10 degrees. The natural

periods can be read from the time duration of one full cycle of the platform motion. The

0 100 200 300 400 500-10

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0

5

10

15

Time(sec)

m

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10

Time(sec)

deg

0 100 200 300 400 500-10

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0

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15

Time(sec)

m

0 100 200 300 400 500-10

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0

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Time(sec)de

g

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Time(sec)

m

0 100 200 300 400 500-10

-5

0

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10

15

Time(sec)

deg

120

natural frequencies, calculated in this free decay test, included the effect of the viscous

drag from the hull and mooring lines as pointed out above.

It is seen that the natural frequencies of surge, sway and yaw, tabulated in Table

7.4, were positioned away from the wave frequency range so that the platform could

avoid the resonance. Heave, roll and pitch natural frequencies were relatively close to

the wave frequency range, but still had enough margins to the peak wave frequency.

Thus, this design of floating system, including three turbines, is acceptable in an

offshore wind-wave environment.

Table 7.4 Natural frequencies of triangular platform

Surge Sway Heave Roll Pitch Yaw

Period (sec) 80.2 79.5 19.4 19.9 20.1 55.7

Freq. (rad/s) 0.08 0.08 0.32 0.32 0.31 0.11

Even though the platform geometry had only one plane symmetry, it was seen

that the natural frequency of surge and sway, or roll and pitch are nearly identical to each

other.

7.4.3 Responses in Random Wind and Wave Environment

So far, the system identification works was done without consideration of

structural elasticity and aero dynamic loading on the blades. It turned out that the current

MUFOWT model can support 3 turbines with proper platform natural frequencies. The

mooring system was also well designed to give a proper restoring force and moment to

121

the floating platform. In this section, a fully working MUFOWT in wind-wave

environment was simulated. For simplicity, the directions of wind and wave were

aligned (collinear) and set to zero degree. Mean wind speed at 90m height, which

corresponds to the hub height of 5MW turbine, was set to 15m/s, which was higher than

the rated wind speed. The environmental conditions are tabulated in Table 7.5.

Table 7.5 Environmental conditions (Triangular platform)

Item Unit Value

Mean Wind Speed At 90m Height m/s 15

Water Depth m 300

Wave Heading deg 0

Significant Wave Height m 5.0

Peak Wave Period sec 8.688

Overshoot parameter - 2.4

Cut-in / Cut-out Wave Frequencies rad/s 0.15 / 1.2

The full field wind data was generated by TurbSim (Jonkman, 2009), and the

wind velocities at different hub heights between 1.5MW and 5MW turbines were

calculated separately. The random waves were generated base on the JONSWAP wave

spectrum with significant wave height of 5m, peak period of 8.688sec, and overshoot

parameter of 2.4. The current was not considered in this case study for simplicity. The

time step of the CHARM3D side, which included the numerical integration of mooring

line equations, was set to 0.01 seconds, and that of the FAST side, which included the

computation of aerodynamics, elastic modes of tower and blades and platform dynamics,

122

was set to 0.005 seconds. So, at every two time step of FAST, the CHARM3D fed the

mooring restoring and all other external loadings on the platform. The total simulation

time was 1,000 seconds, including the initial 100 seconds of ramp time, in order to

minimize the transient effect of responses in the beginning.

(a) (b) Figure 7.8 Surge motion (a) and spectrum (b) (Triangular platform)

(a) (b) Figure 7.9 Sway motion (a) and spectrum (b) (Triangular platform)

The surge and sway responses in Figures 7.8 ~ 7.9 show that the floating

platform motions were primarily dependent on the low frequency, which was derived by

wind. The contribution of wave was relatively small.

0 200 400 600 800 1000-5

0

5

10

Time(sec)

m

0 0.5 1 1.5 20

2

4

6

8

(rad/sec)S

( )

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-0.2

0

0.2

0.4

Time(sec)

m

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

(rad/sec)

S(

)

123

(a) (b) Figure 7.10 Heave motion (a) and spectrum (b) (Triangular platform) In the case of heave, the platform response was primarily affected by wave

energy as can be seen in Figure 7.10. The roll and pitch responses in Figures 7.11 ~ 7.12

show a high peak around its natural frequency of 0.32 rad/s. Since the wave energy in

this frequency range was not significant, no severe resonances occurred for both roll and

pitch motions.

(a) (b) Figure 7.11 Roll motion (a) and spectrum (b) (Triangular platform)

0 200 400 600 800 1000-0.5

0

0.5

1

Time(sec)

m

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

(rad/sec)

S(

)

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0

0.1

0.2

0.3

0.4

Time(sec)

deg

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

(rad/sec)

S(

)

124

(a) (b) Figure 7.12 Pitch motion (a) and spectrum (b) (Triangular platform)

(a) (b) Figure 7.13 Yaw motion (a) and spectrum (b) (Triangular platform)

Figure 7.13 shows that the mean yaw angle was not zero. This yaw offset also

could be seen in a single floating wind turbine due to the tower base torsional moment

which was induced by a rotating inertia of the rotor and a gyroscopic effect. In the case

of a multiple turbines platform, the total torsional moment becomes bigger, so the

resultant yaw response will be significantly increased. So, it is important to design a

floating platform of MUFOWT considering the appreciable yaw moment from all

turbines.

0 200 400 600 800 1000-2

0

2

4

6

Time(sec)

deg

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

(rad/sec)

S(

)

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-0.4

-0.2

0

0.2

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Time(sec)

deg

0 0.5 1 1.5 20

0.05

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0.2

0.25

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S(

)

125

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.14 Generator power (a) and spectra (b) (Triangular platform)

In addition to the platform responses, the turbine outputs were also checked and

presented. The electric power output in Figure 7.14 shows that turbine #1 generated

rated power of 5MW, and the other turbines normally generated 1.5MW power during

the entire simulation time.

Figure 7.15 shows the variation in blade pitch angle. It is seen that the variation

of the 1.5MW blade angle was more active because the rated rotor speed was much

faster than that of the 5MW turbine. Overall, it was apparent that blade pitch variation

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8000

Time(sec)

kW

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S(

)

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2000

4000

6000

8000

Time(sec)

kW

0 0.5 1 1.5 20

2000

4000

6000

(rad/sec)

S(

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2000

4000

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8000

Time(sec)

kW

0 0.5 1 1.5 20

2000

4000

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(rad/sec)

S(

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126

was mostly dependent on the low frequency excitation from wind and a minor

contribution from wave energy.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.15 Blade pitch angle (a) and spectra (b) (Triangular platform)

The blade pitch and the generator torque controller used in this three-turbine case

followed the same method that Hywind spar turbine had adopted. Otherwise, the

platform responses as well as the turbine performance could experience similar

instability observed in Hywind spar case because the pitch natural frequency of this

semi-submersible platform was also very low, and there was possibility of negative

0 200 400 600 800 10000

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20

Time(sec)

deg

0 0.5 1 1.5 20

10

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40

(rad/sec)

S(

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Time(sec)

deg

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40

(rad/sec)

S(

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S(

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127

damping due to the conventional blade pitch controller. For convenience, the controller

gains were adopted from the recommended values by NREL (Jonkman, 2008).

Since the structural specifications were different between the 1.5MW and 5MW

turbines, the tower base loads were also different. As can be seen in Figure 7.16, the

variation and magnitude of the 5MW tower base load was much greater than that of the

1.5MW turbine.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.16 Tower base fore-aft shear force (a) and spectra (b)

(Triangular platform)

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

0 0.5 1 1.5 20

1

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3

4x 10

5

(rad/sec)

S(

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0

500

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1500

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Time(sec)

kN

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S(

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S(

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128

Thus, proper structural design at the tower base becomes essential to ensure

structural integrity. In the case of the tower base in-line shear force, the responses were

very sensitive to the wave energy frequency range.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.17 Tower base axial force (a) and spectra (b) (Triangular platform)

The tower base axial force in Figure 7.17 shows the difference in tower weight

between 1.5MW and 5MW unit. Similarly, the axial forces are also dependent on the

wave frequency range. The standard deviation of axial force from the 5MW turbine is

121.8 kN, and that of 1.5MW turbine is 24.2 kN, which means the repetitive vertical

0 200 400 600 800 1000

-6000

-4000

-2000

0

Time(sec)

kN

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

0 200 400 600 800 1000

-6000

-4000

-2000

0

Time(sec)

kN

0 0.5 1 1.5 20

500

1000

1500

2000

(rad/sec)

S(

)

0 200 400 600 800 1000

-6000

-4000

-2000

0

Time(sec)

kN

0 0.5 1 1.5 20

500

1000

1500

2000

(rad/sec)

S(

)

129

loading on the 5MW tower base is more severe than that of the 1.5MW turbine; thus the

fatigue failure of tower could become an issue for a large scale turbine.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.18 Tower base pitch moment (a) and spectra (b) (Triangular platform)

The tower base fore-aft moments and its spectra are depicted in Figure 7.18. It

shows that the fore-aft moments at the tower base were affected by wave energy, similar

to the fore-aft shear forces.

Due to the scale effect, the tower base torsional moment of the 5MW turbine was

much greater than that of the 1.5MW turbine as seen in Figure 7.19. Also, the tower base

0 200 400 600 800 1000-5

0

5

10

15x 10

4

Time(sec)

kN.m

0 0.5 1 1.5 20

1

2

3x 10

9

(rad/sec)

S(

)

0 200 400 600 800 1000-2

0

2

4x 10

4

Time(sec)

kN.m

0 0.5 1 1.5 20

5

10

15x 10

7

(rad/sec)

S(

)

0 200 400 600 800 1000-2

0

2

4

6x 10

4

Time(sec)

kN.m

0 0.5 1 1.5 20

5

10

15x 10

7

(rad/sec)

S(

)

130

of the 5MW turbine was more vulnerable to torsional fatigue failure compared to the

1.5MW turbine due to the severe repetitive loads from the upper turbine. Interestingly,

the tower base torsional moment was more affected by low frequency excitation from

the wind, while the shear or axial force was primarily affected by wave loadings.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.19 Tower base torsional moment (a) and spectra (b) (Triangular platform)

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

1

2

3x 10

6

(rad/sec)

S(

)

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

131

7.5 One Turbine Failure Simulation

7.5.1 Blade Pitch Control Failure of One Turbine

In this section, a more rigorous and sophisticated simulation of MUFOWT with

the failure of one turbine was carried out and the platform responses, as well as turbine

performances, were checked. The failure of the turbine was implemented by locking the

blade pitch angle at 30 degrees for the smaller turbine #2 in Figure 7.20 (a). In detail, the

blade pitch angle of turbine #2 suddenly went out of control, started to increase at 500

seconds, and then stopped in one minute. The blade pitch angle was locked at 30 degrees,

which was insufficient to rotate the blades.

(a) (b) Figure 7.20 Turbine location (a) and turbine failure (b) (Triangular platform)

Turbine #1

Turbine #2

Turbine #3

Failure

132

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.21 Blade pitch angle (a) and spectra (b) with blade control failure

(Triangular platform) During the first 500 seconds, the simulation performed normally. All turbines

and platform were well balanced and worked without any disturbance. After 500 seconds,

turbine #2 lost its thrust force due to the decreased angle of attack of the blades in

turbine #2. As a result, the floating platform may experience unexpected yaw moment as

seen in Figure 7.20 (b). Figure 7.21 shows the change of blade pitch angle of each

turbine; it is seen that the pitch angle suddenly increases up to 30 degrees at 500 seconds

and locked turbine #2.

0 200 400 600 800 10000

10

20

30

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

40

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

10

20

30

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

2000

4000

6000

8000

10000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

10

20

30

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

20

40

60

(rad/sec)

S(

)

Normal

Fault

133

Sur

ge

Sw

ay

Hea

ve

(a) (b) Figure 7.22 Platform translational motions (a) and spectra (b) with blade control

failure (Triangular platform)

Due to the failure of turbine #2, the overall aerodynamic drag forces from each

turbine were reduced; thus the mean surge offset after 500 seconds also decreased from

5.5m to 4.8m. As can be seen in Figure 7.22, the variation of heave is relatively small

compared to the other modes.

In Figure 7.23, the platform yaw response shows very rapid change from -0.11

degree to +1.23 degree in a few minutes. The amount of yaw angle change seems

relatively small, but this small variation usually induced a very large translational offset

0 200 400 600 800 1000-5

0

5

10

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

2

4

6

8

10

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-0.4

-0.2

0

0.2

0.4

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-0.5

0

0.5

1

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

(rad/sec)

S(

)

Normal

Fault

134

at the turbine location because the MUFOWT platform was bigger than the single

turbine platform.

Rol

l P

itch

Y

aw

(a) (b) Figure 7.23 Platform rotational motions (a) and spectra (b) with blade control


0 200 400 600 800 1000-0.1

0

0.1

0.2

0.3

0.4

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

0.005

0.01

0.015

0.02

0.025

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4

6

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1

-0.5

0

0.5

1

1.5

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

1

2

3

4

(rad/sec)

S(

)

Normal

Fault

135

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.24 Generator power (a) and spectra (b) with blade control failure

(Triangular platform) As for the electric power output in Figure 7.24, turbine #1 produced 5MW rated

power normally, while turbine #2 produced very low electric power around 500 kW after

the locking of the blade pitch angle. This sudden drop in electric power was mainly due

to the slow rotation of rotor due to the feathered blade pitch angle.

The responses of tower base force and moment in Figure 7.25 were also checked

as below. As expected, the fore-aft shear forces at the tower base for turbine #2

decreased appreciably. For example, the mean shear force of turbine #2 dropped from

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

1

2

3

4x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

5

10

15x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

2000

4000

6000

(rad/sec)

S(

)

Normal

Fault

136

185.0 kN to 77.7 kN after the blade pitch failure. This drop of fore-aft shear force is

main reason for the unbalanced yaw loading on the platform.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.25 Tower base fore-aft shear force (a) and spectra (b) with blade control


In the frequency domain, it was seen that after failure at 500 seconds, the low

frequency response was dominant for the tower base fore-aft shear force. It can be

explained that the blade pitch angle was fixed; after that time, the turbine blade could not

control the inflow aero loading and was fully affected by random wind which made a

very low frequency response of the tower base shear force of turbine #2.

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1

2

3

4x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-200

0

200

400

600

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10

15x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-200

0

200

400

600

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1

2

3x 10

4

(rad/sec)

S(

)

Normal

Fault

137

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.26 Tower base pitch moment (a) and spectra (b) with blade control failure

(Triangular platform)

Similar phenomena can be also seen in the tower base pitch moment in Figure

7.26 and tower base torsional moment in Figure 7.27. The mean tower base pitch

moment of turbine #2 decreased by 68.1% due to the loss of drag force. The maximum

tower base torsional moment also decreased by 25.9% after locking the blade pitch angle.

The changes in tower base loadings from other turbines (#1 and #3) were not that

significant.

0 200 400 600 800 1000-5

0

5

10

15x 10

4

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

1

2

3x 10

9

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4x 10

4

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

5

10x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4

6x 10

4

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

5

10

15x 10

7

(rad/sec)

S(

)

Normal

Fault

138

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.27 Tower base torsional moment (a) and spectra (b) with blade control


The failure of turbines also affects the mooring line top tensions. To see the time

series of top tensions and its spectrum, each line is numbered as seen in Figure 7.28. In

case of lee-side mooring lines such as #1 and #2 in Figure 7.29, the top tensions after

failure event are increased while the weather-side lines including #4 and #5 in Figure

7.30 show less top tension after the failure of turbines. This change is mainly due to the

variation of total thrust force on the platform and reduced surge offset. Side lines (#3 and

#6) show relatively less change compared to the other lines because those lines are not

directly related to the platform surge drift.

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

1

2

3x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

4

(rad/sec)

S(

)

Normal

Fault

139

Figure 7.28 Top view of mooring-line arrangement (Triangular platform)

Lin

e #

1 L

ine

#2

(a) (b) Figure 7.29 Top-tension (a) and spectra (b) of Line #1~#3 (Triangular platform)

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5000

10000

15000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

Line #1

Line #2

Line #3

Line #6

Line #4

Line #5

WIND WAVE

140

Lin

e #

3

(a) (b) Figure 7.29 Continued

Lin

e #

4 L

ine

#5

Lin

e #

6

(a) (b) Figure 7.30 Top-tension (a) and spectra (b) of Line #4~#6 (Triangular platform)

The overall performances of the three-turbine semi-submersible has been

numerically computed and presented in this section. The time domain responses,

0 200 400 600 800 10001050

1100

1150

1200

1250

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

500

1000

1500

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001100

1200

1300

1400

1500

1600

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1

2

3

4x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001100

1200

1300

1400

1500

1600

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001050

1100

1150

1200

1250

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

500

1000

1500

(rad/sec)

S(

)

Normal

Fault

141

including aero-elastic-hydro-mooring dynamics for multiple turbines, can be analyzed

through the developed tool. All the turbine outputs and platform responses have been

solved simultaneously, so every possible wind turbine dynamic and interaction between

each turbine and platform can be captured without any time consuming and complicated

derivations.

7.5.2 Partial Blade Broken Failure of One Turbine

The NREL’s 5MW baseline wind turbine has three blades. The blade structural

model is based on the structural properties of the 62.6m-long LM Glassfiber blade used

in the DOWEC study and properly modified by NREL (Jonkman, 2007). The overall

blade mass is 17,740 kg per blade, and the structural damping ratio is 0.477% in all

modes of the isolated blade. Table 7.6 summarizes the blade structural properties.

Table 7.6 Blade structural properties

Item Unit Value

Length (w.r.t. Root Along Preconed Axis) m 61.5

Mass Scaling Factor % 4.536

Overall (Integrated) Mass kg 17,740

Second Mass moment of Inertia (w.r.t. Root) kg·m2 11,776,047

First Mass Moment of Inertia (w.r.t. Root) kg·m 363,231

CM Location (w.r.t. Root Along Preconed Axis) m 20.475

Structural Damping Ratio (All Modes) % 0.477465

142

Similar to the blade structural properties, the blade aerodynamic properties are

also defined based on the DOWEC blades. The details are explained by J. Jonkman

(Jonkman, 2007).

The developed time-domain tool for wind turbine-floater-tether coupled dynamic

analysis has been used in this study to assess the transient, global, and local effects when

a blade tip is suddenly damaged. The details of tower/blade properties and control

schemes are the same as those of NREL’s 5MW Baseline wind turbine. The wind speed

at hub height is set to 15m/s with vertical or horizontal variations, i.e. full-field wind

data is used in this study considering random variation along both horizontal and vertical

directions inside the blade swept area. The blade breaking zones were determined based

on the blade node number. To minimize the instability of the simulation, the blade

breaking zones were selected from the node number 15 to 17. Table 7.7 shows the

selected blade length and node number.

Table 7.7 Damaged blade length and node number

Node number Location from the Apex Element Length Element Mass

15 56.1667 m 2.7334 m 248.21 kg

16 58.9000 m 2.7332 m 193.63 kg

17 61.6333 m 2.7334 m 132.17 kg

143

Figure 7.31 Blade length and broken zone The total simulation time was set to 1,000 seconds, and the breaking event

occurred at 500 seconds of time frame. To minimize the transient responses of the floater

in the beginning, the wave loading was gradually increased to the actual values during

the ramping time of 100s. However, ramping was not applied to wind loading. Instead,

the blade was initially rotated with the given wind loading and thus the initial transient

effect associated with the wind loading on the blade was very minimal. The statistics

were calculated after the ramping period (100 seconds).

To simulate the broken blade, both the aerodynamic and structural dynamic

properties were changed at a given breaking time, i.e. the blade element mass and

aerodynamic forces in the broken range were set to zeros at the time of the breaking

61.5m

8.2m

Lea

ding

edg

e

Tra

ilin

g ed

ge

Blade root

Hub

Blade tip

144

event. Figure 7.31 shows the broken zone of the blade. The total loss of blade element

mass was equivalent to 574 kg, and the length was about 8m.

The simulation result showed that the platform responses are not significantly

influenced by the partially broken blade because the semi-submersible platform was very

compliant, and the mass scale between turbine and platform was quite different. In the

case of the single turbine TLP-type FOWT, the platform responses, especially for roll,

pitch and yaw motions, were appreciably influenced by the partially broken blade.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.32 Tower base side-to-side shear force (a) and spectra (b) with partially

broken blade

0 200 400 600 800 1000-200

-100

0

100

200

300

Time(sec)

kN

Normal

Fault

0 1 2 3 40

2000

4000

6000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-100

-50

0

50

100

Time(sec)

kN

Normal

Fault

0 1 2 3 40

50

100

150

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-100

-50

0

50

100

Time(sec)

kN

Normal

Fault

0 1 2 3 40

50

100

150

(rad/sec)

S(

)

Normal

Fault

145

However, local structural responses such as tower base loads were influenced by

the broken blade as presented in Figures 7.32 ~ 7.34.

Since the blade from the turbine #1 was broken during the simulation, the

imbalance load on the rotor induced a 1P (1.27 rad/s) frequency response as can be seen

in Figure 7.32. Interestingly, that 1P frequency response also showed up in turbines #2

and #3. For turbines #2 and #3; this 1P response was solely from turbine #1 because the

rotor speed that generated the 1P frequency of turbine #1 was different from the other

turbines. That is, the 1P frequency of turbines #2 and #3 was around 2.15 rad/s, which is

also shown in the responses.

Table 7.8 Tower base side-to-side shear force statistics with partially broken blade

Max. Min Mean SD

Turbine #1 (kN)

Normal 4.34E+01 -9.10E+01 -2.70E+01 2.11E+01

Partially Broken 1.23E+02 -1.53E+02 -2.65E+01 4.50E+01

Turbine #2 (kN)

Normal 1.28E+01 -2.90E+01 -7.10E+00 6.55E+00


Turbine #3 (kN)

Normal 1.37E+01 -2.84E+01 -7.34E+00 6.32E+00


In Table 7.8, the maximum shear force and standard deviation of turbine #1 after

breaking increased by 182.8% and 113.0% respectively. Furthermore, the maximum

forces of turbines #2 and #3 also increased by 93.8% ~ 108.6%. This result indicated that

the partially broken blade of one turbine strongly affected the responses of other turbines.

The statistics in Table 7.8 were obtained after breaking event at 500 seconds.

146

Tower base roll moments in Figure 7.33 also showed a similar trend with shear

forces. As mentioned above, both 1.27 rad/s (1P of turbine #1) and 2.15rad/s (1P of

turbines #2 and #3) showed up simultaneously so the structural properties of the tower

base in the MUFOWT platform should be carefully designed by considering every

possible response from the other turbines. The lowest tower bending mode of turbine #1

shows up near 2.2 rad/s, and this response can be also detected in the other turbines such

as #2 and #3. This result confirmed that the dynamic coupling between each turbine was

successfully implemented in the developed program for the MUFOWT analysis.

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.33 Tower base roll moment (a) and spectra (b) with partially broken blade

0 200 400 600 800 1000-1

0

1

2x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6x 10

7

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6

8x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6

8x 10

5

(rad/sec)

S(

)

Normal

Fault

147

Table 7.9 Tower base roll moment statistics with partially broken blade

Max. Min Mean SD

Turbine #1 (kN·m)

Normal 1.11E+04 1.07E+03 6.43E+03 1.69E+03

Partially Broken 1.75E+04 -7.48E+03 6.39E+03 4.28E+03

Turbine #2 (kN·m)

Normal 2.70E+03 -3.41E+02 1.15E+03 4.77E+02


Turbine #3 (kN·m)

Normal 2.65E+03 -3.50E+02 1.17E+03 4.61E+02


Similarly, the maximum roll moment and standard deviation of normal turbines

(#2 and #3) in Table 7.9 also showed appreciable increases after the braking event at 500

seconds.

Tur

bine

#1

Tur

bine

#2

(a) (b) Figure 7.34 Tower base pitch moment (a) and spectra (b) with partially broken

blade

0 200 400 600 800 1000-5

0

5

10

15x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

1

2

3x 10

9

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

5

10

15x 10

7

(rad/sec)

S(

)

Normal

Fault

148

Tur

bine

#3


The variations of tower base pitch moment in Figure 7.34 were relatively small.

The statistics in Table 7.10 showed that the standard deviations of every tower base

moment increased only 0.4 ~ 0.6%, while the maximum moments of all turbines

decreased by 0.8 ~ 1.3%. In fore-aft direction, the reduced blades drag due to the loss of

partial elements resulted in these decreases. It revealed that the loss of partial blade

element did not make significant differences in the tower base fore-aft (pitch) moment.

Table 7.10 Tower base pitch moment statistics with partially broken blade

Max. Min Mean SD

Turbine #1 (kN·m)

Normal 1.13E+05 -1.18E+04 5.16E+04 2.25E+04


Turbine #2 (kN·m)

Normal 2.93E+04 -3.18E+03 1.32E+04 5.19E+03


Turbine #3 (kN·m)

Normal 2.94E+04 -1.42E+03 1.32E+04 5.21E+03


0 200 400 600 800 1000-2

0

2

4

6x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

5

10

15x 10

7

(rad/sec)

S(

)

Normal

Fault

149

The 1P response from the broken blade was also shown in the tower base

torsional moment response as can be seen in Figure 7.35. However, that 1P did not show

up in turbines #2 and #3 anymore because the yaw excitation of turbine #1 generated

most of the sway or roll excitations for the other turbines, and contributed negligibly

small yaw excitations. The statistics of tower base torsional moment in Table 7.11 also

show that the variations of moment from turbines #2 and #3 were not that great and only

those of turbine #1 increased noticeably. For example, the maximum torsional moment

and standard deviation increased by 32.3% and 58.7%. It tells us that the modes of

lateral direction such as sway and roll are primarily influenced by the imbalance of rotor

blades. To investigate more severe breaking event and the resultant coupling effects,

fully broken blade case are presented in the next section.

Table 7.11 Tower base torsional moment statistics with partially broken blade

Max. Min Mean SD

Turbine #1 (kN·m)

Normal 3.00E+03 -3.29E+03 -1.54E+02 9.25E+02


Turbine #2 (kN·m)

Normal 5.17E+02 -6.45E+02 -5.00E+01 1.55E+02


Turbine #3 (kN·m)

Normal 4.41E+02 -6.44E+02 -7.19E+01 1.48E+02


150

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.35 Tower base torsional moment (a) and spectra (b) with partially broken

blade The mooring line top tensions in this partially broken blade case were also

investigated and presented in Figures 7.36 ~ 7.37. The mooring line arrangement was

depicted in Figure 7.28.

0 200 400 600 800 1000-6000

-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6x 10

4

(rad/sec)

S(

)

Normal

Fault

151

Lin

e #

1 L

ine

#2

Lin

e #

3

(a) (b) Figure 7.36 Top-tension (a) and spectra (b) of Line #1~#3 with partially broken

blade Except for the minor 1P response in line #3 tensions, the top tension responses

did not make significant differences between before and after the breaking event. The

statistics of top tensions were obtained after 500 seconds and tabulated in Table 7.12.

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001050

1100

1150

1200

1250

Time(sec)

kN

Normal

Fault

0 1 2 3 40

500

1000

1500

(rad/sec)

S(

)

Normal

Fault

152

Lin

e #

4 L

ine

#5

Lin

e #

6

(a) (b) Figure 7.37 Top-tension (a) and spectra (b) of Line #4~#6 with partially broken

blade

In the case of the other lines in Figure 7.37, the trends were similar to that of

Figure 7.36. The statistics in Table 7.12 showed that there were no significant changes in

top tensions in partially broken blade case. The standard deviations of side lines (#3 and

#6) increased only by 0.9~ 1% and that of the other lines increased less than 0.5%. Thus,

it was regarded that the effects of partially broken blade on the mooring line top tensions

in this semi-submersible platform were minor compared to the side-to-side tower base

responses.

0 200 400 600 800 10001100

1200

1300

1400

1500

1600

Time(sec)

kN

Normal

Fault

0 1 2 3 40

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001100

1200

1300

1400

1500

1600

Time(sec)

kN

Normal

Fault

0 1 2 3 40

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001050

1100

1150

1200

1250

Time(sec)

kN

Normal

Fault

0 1 2 3 40

500

1000

1500

(rad/sec)

S(

)

Normal

Fault

153

Table 7.12 Mooring line top tension statistics with partially broken blade

Max. Min Mean SD

Line #1 (kN)

Normal 1.04E+03 8.60E+02 9.49E+02 3.04E+01

Partially Broken 1.04E+03 8.59E+02 9.49E+02 3.04E+01

Line #2 (kN)

Normal 1.04E+03 8.60E+02 9.49E+02 3.04E+01


Line #3 (kN)

Normal 1.21E+03 1.09E+03 1.15E+03 2.22E+01


Line #4 (kN)

Normal 1.57E+03 1.16E+03 1.37E+03 6.48E+01


Line #5 (kN)

Normal 1.55E+03 1.13E+03 1.36E+03 6.50E+01


Line #6 (kN)

Normal 1.21E+03 1.08E+03 1.15E+03 2.22E+01


7.5.2 Full Blade Broken Failure of One Turbine

In this section, more severe failure case with 100% loss of the one blade was

simulated and assessed. All the conditions were same as the previous partially broken

case but the entire elements of one blade were eliminated at 500 seconds. Consequently,

the aero dynamic loadings on that blade also should be removed. The overall mass

removed from the broken blade was equivalent to 17,740 kg. Compared to the partially

broken case, the platform and turbine responses considerably increased due to the

imbalanced blade mass and unbalanced excitation of the aero dynamic loadings.

154

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.38 Tower base fore-aft shear force (a) and spectra (b) with fully broken

blade Interestingly, the tower base fore-aft shear forces after breaking showed clear

coupling between turbines. The 1P excitation of the turbine #1 in Figure 7.38 could be

also seen in turbines #2 and #3, and two lowest tower bending modes around 2 ~ 3 rad/s

also showed up in each turbine.

Table 7.13 showed that the maximum fore-aft shear force of the turbine #1

increased nearly 17.3%. Another turbine’s maximum also increased by 16.9 ~ 20.1%. In

the case of a partially broken blade in the previous section, the variation of a fore-aft

0 200 400 600 800 1000-1000

0

1000

2000

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1

2

3

4x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-200

0

200

400

600

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1

2

3x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-200

0

200

400

600

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1

2

3x 10

4

(rad/sec)

S(

)

Normal

Fault

155

shear force was not noticeable, whereas the excessive rotation of the unbalanced-blade in

this case could induce even for the in-line (fore-aft) shear force variations.

Table 7.13 Tower base fore-aft shear force statistics with fully broken blade

Max. Min Mean SD

Turbine #1 (kN)

Normal 1.38E+03 -2.19E+02 6.07E+02 2.81E+02

Fully Broken 1.62E+03 -5.22E+02 6.00E+02 3.54E+02

Turbine #2 (kN)

Normal 4.23E+02 -6.05E+01 1.85E+02 7.68E+01


Turbine #3 (kN)

Normal 4.18E+02 -3.70E+01 1.85E+02 7.71E+01


As already expected, the tower base side-to-side shear forces as well as roll

moments showed great changes after 500 seconds. The tower base frequency responses

in Figure 7.39(b) indicated that the unbalanced vibration responses were composed of

both 1P excitations from the blades and the tower base side-to-side bending modes from

two turbines. The most dominant component was the lowest bending mode of the 5MW

tower near the 2.2 rad/s. The statistics in Table 7.14 shows the considerable increase of

the shear forces. In real situations, these changes might result in the progressive collapse

of the tower and the entire system.

156

Table 7.14 Tower base side-to-side shear force statistics with fully broken blade

Max. Min Mean SD

Turbine #1 (kN)

Normal 4.34E+01 -9.10E+01 -2.70E+01 2.11E+01

Fully Broken 2.33E+03 -2.28E+03 -2.84E+01 7.18E+02

Turbine #2 (kN)

Normal 1.28E+01 -2.90E+01 -7.10E+00 6.55E+00


Turbine #3 (kN)

Normal 1.37E+01 -2.84E+01 -7.34E+00 6.32E+00


Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.39 Tower base side-to-side shear force (a) and spectra (b) with fully

broken blade

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN

Normal

Fault

0 1 2 3 40

5

10

15x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN

Normal

Fault

0 1 2 3 40

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN

Normal

Fault

0 1 2 3 40

5

10

15x 10

4

(rad/sec)

S(

)

Normal

Fault

157

The changes in the tower base pitch moment in Figure 7.40 were very similar to

the fore-aft shear forces in Figure 7.38. Again, it was very hard to observe the variations

in the tower base pitch moment for the partially broken blade case, but in this fully

broken case, clear effects could be observed. The 1P response of the 5MW turbine and

two tower base bending modes were dominant components in this simulation.

Table 7.15 shows that the maximum tower base pitch moment increased up to

33.2%, and the standard deviation of the moment also increased by 59.0%. The

maximum of another turbines also increased by 22.9 ~ 25.2%. Under the blade-broken

situations, the tower base and turbine structure could be exposed to higher possibility of

fatigue failure.

Table 7.15 Tower base pitch moment statistics with fully broken blade

Max. Min Mean SD

Turbine #1 (kN·m)

Normal 1.13E+05 -1.18E+04 5.16E+04 2.25E+04


Turbine #2 (kN·m)

Normal 2.93E+04 -3.18E+03 1.32E+04 5.19E+03


Turbine #3 (kN·m)

Normal 2.94E+04 -1.42E+03 1.32E+04 5.21E+03


158

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.40 Tower base pitch moment (a) and spectra (b) with fully broken blade

In the case of tower base torsional moment under the full blade breaking, the

maximum torsional moment of the turbine #1 was around 5.7 times greater than that of

the normal case as can be seen in Figure 7.41 and Table 7.16. The responses of the

turbine #1 were the most serious changes, while other turbines showed relatively mild

changes. If the structural safety factor of the tower base design was not enough to cover

those variations, then the serious failure of the turbines were expected.

0 200 400 600 800 1000-1

0

1

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

1

2

3x 10

9

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

5

10

15x 10

7

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

0

2

4

6x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

5

10

15x 10

7

(rad/sec)

S(

)

Normal

Fault

159

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

(a) (b) Figure 7.41 Tower base torsional moment (a) and spectra (b) with fully broken

blade One of the interesting responses in torsional moment was that the tower base

side-to-side excitation of turbine #1 could be observed in the tower base torsional

responses in turbines #2 and #3. Since the tower center lines were not necessarily located

at the platform origin in the case of MUFOWT, the linear excitation from one tower base

might result in the angular responses in the other tower base and vice versa as can be

seen in Figure 7.42.

0 200 400 600 800 1000-2

-1

0

1

2x 10

4

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

1

2

3

4x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-1000

-500

0

500

1000

Time(sec)

kN.m

Normal

Fault

0 1 2 3 40

2

4

6

8x 10

4

(rad/sec)

S(

)

Normal

Fault

160

Figure 7.42 Transition of the side-to-side excitation of turbine #1

Table 7.16 Tower base torsional moment statistics with fully broken blade

Max. Min Mean SD

Turbine #1 (kN·m)

Normal 3.00E+03 -3.29E+03 -1.54E+02 9.25E+02


Turbine #2 (kN·m)

Normal 5.17E+02 -6.45E+02 -5.00E+01 1.55E+02


Turbine #3 (kN·m)

Normal 4.41E+02 -6.44E+02 -7.19E+01 1.48E+02


The mooring line top tensions in this emergency case were also investigated and

presented in Figures 7.43 ~ 7.44.

1P

1P

161

Lin

e #

1 L

ine

#2

Lin

e #

3

(a) (b) Figure 7.43 Top-tension (a) and spectra (b) of Line #1~#3 with fully broken blade

Two lee-side lines #1 and #2 did not make significant differences and the only

minor responses in 1P frequency showed up. However, the top tensions of lines #3 and

#6 which was located at the side direction showed noticeable changes. The 1P excitation

was the primary source of the excitation from the rotor and the rest of the responses from

the tower base bending modes were relatively small. The maximum top tension of the

line #3 increased by only 3.7%, but the standard deviation after breaking event was

101.1% higher than that of the normal case. Thus, it was regarded that the mooring lines

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000800

900

1000

1100

1200

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001000

1100

1200

1300

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

5000

(rad/sec)

S(

)

Normal

Fault

162

under this situation were more vulnerable to the fatigue failures rather than the breaking

failures from the axial tensions.

Lin

e #

4 L

ine

#5

Lin

e #

6

(a) (b) Figure 7.44 Top-tension (a) and spectra (b) of Line #4~#6 with fully broken blade

In the case of weather-side lines such as lines #4 and #5, the 1P responses in

tension was relatively small compared to the side lines, but still more clear than those of

the lee-side lines. Similarly, the standard deviations of those weather-side lines after

breaking increased around 21.5 ~ 24.5%, which was still larger than the increased rate of

0 200 400 600 800 10001000

1200

1400

1600

1800

Time(sec)

kN

Normal

Fault

0 1 2 3 40

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001100

1200

1300

1400

1500

1600

Time(sec)

kN

Normal

Fault

0 1 2 3 40

0.5

1

1.5

2x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001050

1100

1150

1200

1250

1300

Time(sec)

kN

Normal

Fault

0 1 2 3 40

1000

2000

3000

4000

(rad/sec)

S(

)

Normal

Fault

163

the maximum tensions. The statistics were obtained after the full breaking at 500

seconds, and tabulated in Table 7.17.

Table 7.17 Mooring line top tension statistics with fully broken blade

Max. Min Mean SD

Line #1 (kN)

Normal 1.04E+03 8.54E+02 9.47E+02 2.96E+01

Fully Broken 1.05E+03 8.39E+02 9.48E+02 3.32E+01

Line #2 (kN)

Normal 1.04E+03 8.60E+02 9.49E+02 3.04E+01


Line #3 (kN)

Normal 1.21E+03 1.09E+03 1.15E+03 2.22E+01


Line #4 (kN)

Normal 1.57E+03 1.16E+03 1.37E+03 6.48E+01


Line #5 (kN)

Normal 1.55E+03 1.13E+03 1.36E+03 6.50E+01


Line #6 (kN)

Normal 1.21E+03 1.08E+03 1.15E+03 2.22E+01


7.6 Discussion

The dynamic responses of the three-turbine MUFOWT has been simulated and

investigated in this chapter. The hydrodynamic coefficients were obtained by WAMIT

and the time domain analysis was carried out using the FAST-CHARM3D analysis tool

specifically designed for multiple-turbine platform.

In this case study, it was seen that the effect due to the partial loss of blade pitch

control or partial loss of blade element can significantly affect the responses of the

164

whole system. Specifically, the loss of blade pitch control of one turbine may induce

platform yaw moment which can seriously impact the other turbines due to the yaw error.

In the case of a partially broken blade, the tower-base forces and moments, especially for

sway and roll directions, were the most serious changes compared to normal responses.

Due to the rotational imbalance with damage, the 1P excitation and responses were more

pronounced in the tower and blade dynamics. Interestingly, the 1P excitation from the

broken turbine may influence the other normal turbines, and vice versa. To avoid

collapse of the entire system due to the partially broken blade, the structural integrity,

especially for the yaw-related responses, should be carefully checked. More severe case

with fully broken blade case was also investigated. Under this environment, the

excessive unbalanced forces generated by rotor could influence not only the side-to-side

responses, but also the fore-aft responses. Furthermore, this case also changed the

mooring line top tensions specifically for the side lines.

The present approach for MUFOWT can directly be applied to the development

of remote structural health monitoring systems to detect partial blade failure by

measuring tower or platform responses.

165

8. CASE STUDY III: FIVE-TURBINE SEMI-SEBMERSIBLE

8.1 Introduction

In the previous chapter, the triangular shape MUFOWT platform with three

5MW turbines were numerically modeled and analyzed. The developed MUFOWT

analysis tool was successfully utilized and demonstrated every aspect of the combined

dynamic behaviors of multiple turbines. The fault scenario with one or two turbine

failures was also simulated. It was shown that the failure of turbines may induce

unexpected yaw moment of the floating turbine resulting in significant changes in the

turbine structural load or mooring line top tensions.

In this case study, a bigger MUFOWT platform with 5 wind turbines was

introduced and simulated. Two 5MW turbines were located at the rear side, and three

5MW turbines positioned at the front. To minimize the static trim angle of the platform,

more ballast water filled the aft tank in a way similar to the previous chapter. To confirm

the dynamic coupling between the turbine and platform, an emergency scenario with a

partially broken blade was presented. All the dynamic responses, including each turbine

and platform, have been presented and analyzed in this chapter.

8.2 Configuration of the Platform, Mooring System and Turbines

The selected MUFOWT platform has five cylindrical columns where each

turbine is mounted plus seven submerged pontoons as can be seen in Figure 8.1. The

column spacing was selected to avoid the interruption between turbines. The horizontal

166

distance from the fore turbine to the rear turbine was set to 140m, and the maximum

distance from the upper and lower turbines was 280m. The column depth was 20m, and

the diameter 10m so that the top side of the column can provide the 5MW tower base

with the proper margins. Figure 8.2 shows the dimensions of the platform.

(a) (b) Figure 8.1 Rectangular platform geometry (a) and system configuration (b)

(a) (b) Figure 8.2 Platform dimensions from top (a) and side (b) (Rectangular platform)

The platform hull weight was calculated based on the submerged surface area

and given plate thickness (15mm). The mass properties of the platform, including mass

167

moment of inertia, were calculated by considering the displaced steel platform and

ballast water. The details of platform properties are tabulated in Table 8.1.

The overall hydrostatic equilibrium was met using a different ballast height as

similarly treated in the three-turbine platform. In the case of the five-turbine platform,

the ballast water height at the rear tank is higher than that of the front tanks because the

turbines mounted on top of the cylinder were the same for all columns and the rear side

has only two turbines.

Table 8.1 Specification of rectangular platform

Item Unit Value


Column Diameter m 10.0

Length between Columns (vertical) m 140.0

Length between Columns (lateral) m 156.52

Pontoon Width m 4.0

Pontoon Height m 4.0

Platform Weight, including Ballast N 164,515,390

Platform Buoyancy N 202,426,451

Platform CM Location Below SWL m 15.1

Platform Roll Inertia about CM kg·m2 1.14479E11


Platform Yaw Inertia about Platform Centerline kg·m2 1.63311E11

168

This five-turbine MUFOWT platform was moored by 4 catenary mooring lines.

The material used in the mooring line was the same as the one used in the previous

chapter. The mooring line arrangements are depicted in Figure 8.3.

Figure 8.3 Mooring line configuration (Rectangular platform)

8.3 Hydrodynamic Coefficients in Frequency Domain

The hydrodynamic coefficients such as added mass, radiation damping and first

order hydrodynamic force were obtained from WAMIT.

Figure 8.4 Discretized panel model of floating body (Rectangular platform)

-50

0

50

-100

-50

0

50

100

-20-10

0

x

y

z

169

Similar to the triangular platform in the previous chapter, the platform has one

axis symmetric, and each side has 3,849 panels. Figure 8.4 shows the discretized panel

model of the rectangular floater. The calculated hydrodynamic added mass and radiation

damping in frequency domain are presented in Figures 8.5 and 8.6 respectively.

Surge Roll

Sway Pitch

Heave Yaw

Figure 8.5 Added mass (Rectangular platform)

0.5 1 1.5 21

1.2

1.4

1.6x 10

7

(rad/sec)

Add

ed M

ass

A11

0.5 1 1.5 28.5

8.6

8.7

8.8x 10

10

(rad/sec)A

dded

Mas

s A

44

0.5 1 1.5 20.8

1

1.2

1.4

1.6x 10

7

(rad/sec)

Add

ed M

ass

A22

0.5 1 1.5 24.25

4.3

4.35

4.4x 10

10

(rad/sec)

Add

ed M

ass

A55

0.5 1 1.5 21.36

1.38

1.4

1.42x 10

7

(rad/sec)

Add

ed M

ass

A33

0.5 1 1.5 21

1.2

1.4

1.6

1.8x 10

11

(rad/sec)

Add

ed M

ass

A66

170

In addition to the added mass and radiation damping, hydrostatic stiffness of the

platform can also be obtained from WAMIT. This type of semi-submersible platform

should have enough hydrostatic restoring force and moment which can be confirmed by

calculated hydrostatic coefficients. Otherwise, the platform in still water may not be

stable or cannot maintain its upright position.

Surge Roll

Sway Pitch

Heave Yaw

Figure 8.6 Radiation damping (Rectangular platform)

0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

Dam

ping

B11

0.5 1 1.5 20

0.5

1

1.5

2x 10

9

(rad/sec)

Dam

ping

B44

0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

Dam

ping

B22

0.5 1 1.5 20

2

4

6

8x 10

8

(rad/sec)

Dam

ping

B55

0.5 1 1.5 20

1

2

3x 10

5

(rad/sec)

Dam

ping

B33

0.5 1 1.5 20

2

4

6

8x 10

10

(rad/sec)

Dam

ping

B66

171

8.4 Time Marching Simulation in Normal Operational Condition

In this section, the time domain analysis with given hydrodynamic data derived

by WAMIT has been carried out. Four higher-order FEM based mooring lines have been

modeled and the dynamics of the mooring lines have also been accounted for. At each

time step, the developed MUFOWT analysis tool based on FAST refers to CHARM3D

once to provide all the external loadings including the mooring line restoring force. By

repeating this data exchange, the progressive time marching solution of entire system

can be obtained.

8.4.1 Viscous Damping Modeling

As already pointed out, one of the advantages of the time domain analysis is that

numerical model can include the nonlinear viscous effect. In FAST-CHARM3D, the

viscous damping is calculated by placing equivalent truss or plate members inside the

submerged platform model where the diameter of the member is small compared to the

wave length.

In the case of the five turbines platform, the viscous drag force of the hull was

represented by employing five vertical truss members for each column which acts along

every normal direction of the column, and seven truss members for each pontoon. The

drag coefficient dC is taken to be 0.6 for the cylindrical. In addition to the hull viscous

members above, the dynamic loadings on tethers were also calculated by Morison’s

equation. For the calculation of the loading on tethers, the inertia coefficient mC was

172

taken to be 2 and the drag coefficients dC were set to 1.3 for the wire and 2.4 for the

chain. The drag coefficients for all the Morison members are tabulated in Table 8.2.

Table 8.2 Drag coefficients of Morison members (Rectangular platform)

Location dC Viscous model No. of members

Columns 0.6 Truss 5

Pontoons 0.6 Truss 7

Chain Mooring 2.4 Slender rod 4

Wire Mooring 1.3 Slender rod 4

8.4.2 System Identification and Free Decay Test

To check the 6-DOFs platform natural frequencies, free decay tests in the time

domain were carried out and presented in Figure 8.7. Similar to the three turbines case,

the whole system was assumed as a rigid body, thus no elastic or aerodynamic responses

were included in this free decay test. However, all the viscous effects from the hull or

mooring lines have been considered and included in the simulation.

Surge Roll

Figure 8.7 Free decay test (Rectangular platform)

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)

m

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)

deg

173

Sway Pitch

Heave Yaw

Figure 8.7 Continued The floating platform was released from the 10m offset distance in case of surge,

sway and heave and a 10 degree angle for roll, pitch and yaw. The natural periods or

frequencies can be estimated by measuring the period of free decay oscillations.

Table 8.3 Natural frequencies of rectangular platform

Surge Sway Heave Roll Pitch Yaw

Period 72.6 151.0 18.1 15.9 17.2 73.1

Freq. (rad/s) 0.09 0.04 0.35 0.40 0.37 0.09

Table 8.3 shows the natural frequencies of 6-DOFs of platform motion. Heave,

roll and pitch natural frequencies are relatively close to the wave energy range, but still

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)

m

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)

deg

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)

m

0 100 200 300 400 500-10

-5

0

5

10

Time(sec)de

g

174

look reasonable in that those are far below the peak wave energy range. In case of surge,

sway and yaw, the natural frequencies are quite low, and the resonances can be safely

avoided.

8.4.3 Responses in Random Wind and Wave Environments

After checking the natural frequencies of the system a fully coupled time domain

analysis of MUFOWT was performed. All the environmental conditions are the same as

those in the previous three turbines case study tabulated in Table 7.5.

The time step of the CHARM3D side in this study was set to 0.01 seconds, and

that of the FAST side, which included the computation of aerodynamics, elastic modes

of tower and blades and platform dynamics, was set to the same time step. So, at every

time step of FAST, the CHARM3D fed the mooring restoring and all other external

loadings on the platform. The total simulation time was 1,000 seconds, including the

initial 100 seconds of ramp time in order to minimize the transient effect of responses.

All 6-DOFs of platform responses are presented in Figures 8.8 ~ 8.13.

(a) (b) Figure 8.8 Surge motion (a) and spectrum (b) (Rectangular platform)

0 200 400 600 800 1000-5

0

5

10

15

Time(sec)

m

0 0.5 1 1.5 20

2

4

6

8

10

(rad/sec)

S(

)

175

(a) (b) Figure 8.9 Sway motion (a) and spectrum (b) (Rectangular platform)

(a) (b) Figure 8.10 Heave motion (a) and spectrum (b) (Rectangular platform)

(a) (b) Figure 8.11 Roll motion (a) and spectrum (b) (Rectangular platform)

0 200 400 600 800 1000-4

-2

0

2

4

Time(sec)

m

0 0.5 1 1.5 20

5

10

15

20

(rad/sec)

S(

)

0 200 400 600 800 1000-0.5

0

0.5

1

1.5

Time(sec)

m

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

(rad/sec)

S(

)

0 200 400 600 800 1000-0.05

0

0.05

0.1

0.15

Time(sec)

deg

0 0.5 1 1.5 20

2

4

6

8x 10

-3

(rad/sec)

S(

)

176

(a) (b) Figure 8.12 Pitch motion (a) and spectrum (b) (Rectangular platform)

(a) (b) Figure 8.13 Yaw motion (a) and spectrum (b) (Rectangular platform) As can be seen in the Figures above, most platform responses are located at

below 0.5 rad/s except for heave and pitch. So, attention should be paid and a clear

check done for heave and pitch to ensure a good performance of the platform in wind

and wave environments.

0 200 400 600 800 1000-2

-1

0

1

2

Time(sec)

deg

0 0.5 1 1.5 20

0.01

0.02

0.03

0.04

0.05

(rad/sec)

S(

)

0 200 400 600 800 1000-1

-0.5

0

0.5

1

Time(sec)

deg

0 0.5 1 1.5 20

0.5

1

1.5

2

(rad/sec)

S(

)

177

Tur

bine

#1

Tur

bine

#3

Tur

bine

#5

(a) (b) Figure 8.14 Generator power (a) and spectra (b) (Rectangular platform)

The performance of each turbine was also investigated. In normal operational

condition, all turbines generate rated power of 5MW; the selected output is depicted in

Figure 8.14. Now that the wind speed at hub height (15 m/s) is higher than the rated

wind speed (11.4 m/s), the power generated in this turbine shows only small variations

around rated power. If the wind velocity is near the rated wind speed, then the generated

power may have serious fluctuations due to lowered wind velocity and reduced rotor

speed.

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

0 0.5 1 1.5 20

2

4

6

8x 10

5

(rad/sec)

S(

)

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

0 0.5 1 1.5 20

2

4

6x 10

5

(rad/sec)

S(

)

0 200 400 600 800 10000

2000

4000

6000

8000

Time(sec)

kW

0 0.5 1 1.5 20

2

4

6x 10

5

(rad/sec)

S(

)

178

Tur

bine

#1

Tur

bine

#3

Tur

bine

#5

(a) (b) Figure 8.15 Blade pitch angle (a) and spectra (b) (Rectangular platform)

The variation of blade pitch angle of the selected turbine is depicted in Figure

8.15. As already mentioned, the control action of the blade pitch angle is very sensitive

to low frequency wind energy. It can be confirmed by the spectra in Figure 8.15(b). To

minimize the instability induced by the blade pitch angle control mentioned in the

previous chapter, the same pitch control scheme, which was used in the Hywind spar

case, was adopted because the platform pitch natural frequency in this platform was also

very low with the possibility of interference from blade pitch control excitation.

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

0 0.5 1 1.5 20

10

20

30

40

(rad/sec)

S(

)

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

0 0.5 1 1.5 20

10

20

30

(rad/sec)

S(

)

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

0 0.5 1 1.5 20

10

20

30

40

(rad/sec)

S(

)

179

As for the tower base loads in the in-line direction (fore-aft direction), the

responses are mostly dependent on the incident wave energy. Figure 8.16 shows the time

history and its spectra of tower base fore-aft shear force. Some fluctuations inside the

wave frequency range in spectra were due to the geometry of the platform and the effect

of trapped water inside the outer hull.

Tur

bine

#1

Tur

bine

#3

Tur

bine

#5

(a) (b) Figure 8.16 Tower base fore-aft shear force (a) and spectra (b)

(Rectangular platform) The tower base fore-aft (pitch) bending moment was also checked in Figure 8.17.

The overall trends of the bending moment were very similar to those of the shear forces

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

180

in Figure 8.16. This showed that the tower base loads in normal operational conditions

were nearly identical each other, thus the balance of the entire system was regarded to be

well maintained. As already pointed out, the frequent drops of the spectra inside the

wave frequency range were primarily due to the geometry of the semi-submersible.

Tur

bine

#1

Tur

bine

#3

Tur

bine

#5

(a) (b) Figure 8.17 Tower base pitch moment (a) and spectra (b) (Rectangular platform)

In the case of the tower base torsional moment in Figure 8.18, the response was

affected by low frequency wind energy. It shows that the structural design or fatigue

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

181

analysis of the local structure of the wind turbine should consider the sensitive frequency

range that each structure is subjected to.

Tur

bine

#1

Tur

bine

#3

Tur

bine

#5

(a) (b) Figure 8.18 Tower base torsional moment (a) and spectra (b)

(Rectangular platform)

8.5 One Turbine Failure Simulation

Similar to the previous chapter, the turbine failure case was selected and

simulated to observe the overall performance of turbines and platform. In the previous

chapter, the blade pitch angle of one turbine went wrong and locked at 30 degrees which

showed the lack of drag force on that turbine and the progressive yaw rotation of the

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

0 0.5 1 1.5 20

1

2

3

4x 10

6

(rad/sec)

S(

)

182

platform was eventually detected. In this chapter, on the contrary, the blade pitch angle

suddenly locked at 5 degrees, which is very small compared to the normal pitch angle

action. In this case, the drag force on the blades will be significantly increased, and the

fault turbine will be pushed backward. If that turbine is located in the centerline of the

platform, then only the surge offset will be changed. However, if the fault turbine is off-

centered, the floating platform may experience both translational force and rotational

moment at the same time.

Figure 8.19 Turbine location (Rectangular platform)

Figure 8.19 shows the location of the five turbines. The fault turbine in this

chapter is turbine #3 located at the upper front side of the platform. All other turbines

remained intact during the whole simulation time of 1,000 seconds.

Turbine #2

Turbine #1

Turbine #3

Turbine #4

Turbine #5

183

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

Tur

bine

#4

Tur

bine

#5

(a) (b) Figure 8.20 Blade pitch angle (a) and spectra (b) with blade control failure


0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

40

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

40

50

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

100

200

300

400

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

40

50

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

5

10

15

20

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

40

(rad/sec)

S(

)

Normal

Fault

184

Figure 8.20 shows the change in the blade pitch angle of each turbine. It is seen

that the pitch angle suddenly decreased 5 degrees at 500 seconds for turbine #3. All

other turbines seemed to work normally even after the failure of turbine #3, but the trend

of the pitch angle maneuvering after 500 seconds was slightly different from those of the

normal case because all other turbines tried to compensate for the changes made by

turbine #3.

Sur

ge

Sw

ay

Hea

ve

(a) (b) Figure 8.21 Platform translational motions (a) and spectra (b) with blade control

failure (Rectangular platform)

0 200 400 600 800 1000-5

0

5

10

15

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

5

10

15

20

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-4

-2

0

2

4

6

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

50

100

150

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-0.5

0

0.5

1

1.5

Time(sec)

m

Normal

Fault

0 0.5 1 1.5 20

0.02

0.04

0.06

(rad/sec)

S(

)

Normal

Fault

185

Figures 8.21 ~ 8.22 show the overall platform responses after the failure event. In

Figure 8.21, the platform sway shows a very rapid change from 0.33m to a maximum

5.38m after the failure event. Aside from the platform sway response, the maximum

surge offset also increased from 9.21m to 10.29m due to the increased drag force from

turbine #3.

Platform yaw changes after the failure event were also noticeable as can be seen

in Figure 8.22. The yaw angle changed from -0.11 degrees at 500 seconds, and reached a

minimum of -2.27 degrees in several minutes.

Rol

l P

itch

Y

aw

(a) (b) Figure 8.22 Platform rotational motions (a) and spectra (b) with blade control


0 200 400 600 800 1000-0.05

0

0.05

0.1

0.15

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

0.002

0.004

0.006

0.008

0.01

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-2

-1

0

1

2

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

0.05

0.1

0.15

0.2

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-3

-2

-1

0

1

Time(sec)

deg

Normal

Fault

0 0.5 1 1.5 20

10

20

30

(rad/sec)

S(

)

Normal

Fault

186

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

Tur

bine

#4

Tur

bine

#5

(a) (b) Figure 8.23 Generator power (a) and spectra (b) with blade control failure


0 200 400 600 800 10000

2000

4000

6000

8000

10000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

2

4

6

8x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

10000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

10000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

10000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

2

4

6

8x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

2000

4000

6000

8000

10000

Time(sec)

kW

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

5

(rad/sec)

S(

)

Normal

Fault

187

As pointed out in the previous chapter, the changes in yaw seem very small,

approximately 2 ~ 3 degrees. However, this amount of yaw angle may induce

appreciable changes in local structural responses. So, the overall balance of the

MUFOWT system in terms of static and dynamic should be carefully checked in the

design stage. Otherwise, a small problem in one turbine may affect the performance of

the other turbines. To address these difficulties in system management, a more advanced

turbine control method is necessary. For example, once unbalance of the platform is

detected then control the blade pitch angle or rotor speed so that the system can

minimize the loss of power production or reduce the local structural loadings.

As for the electric power output in Figure 8.23, all turbines except for #3

normally produced 5MW rated power during the entire simulation time, while turbine #3

produced very high electric power up to 10MW after locking the blade pitch angle. This

change in electric power was primarily due to the fast rotation of the rotor due to the

small blade pitch angle. However, this variation in electric power is unrealistic, and

usually controlled by the shaft brake or tip brake. Otherwise, mechanical or electrical

damage may occur and the entire system will be seriously damaged or may collapse.

The responses of the tower base force and moment were also checked as shown

in Figures 8.24 ~ 8.26. As expected, the fore-aft shear forces and bending moments at

the tower base in Figures 8.24 and 8.25 for turbine #3 increased appreciably due to the

increased drag from the blades.

188

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

Tur

bine

#4

Tur

bine

#5

(a) (b) Figure 8.24 Tower base fore-aft shear force (a) and spectra (b) with blade control


0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-500

0

500

1000

1500

2000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10x 10

4

(rad/sec)

S(

)

Normal

Fault

189

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

Tur

bine

#4

Tur

bine

#5

(a) (b) Figure 8.25 Tower base pitch moment (a) and spectra (b) with blade control failure


0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

0.5

1

1.5

2x 10

10

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10000

0.5

1

1.5

2x 10

5

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

8

(rad/sec)

S(

)

Normal

Fault

190

Tur

bine

#1

Tur

bine

#2

Tur

bine

#3

Tur

bine

#4

Tur

bine

#5

(a) (b) Figure 8.26 Tower base torsional moment (a) and spectra (b) with blade control


0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 1000-4000

-2000

0

2000

4000

Time(sec)

kN.m

Normal

Fault

0 0.5 1 1.5 20

2

4

6x 10

6

(rad/sec)

S(

)

Normal

Fault

191

For example, the mean tower base shear force of turbine #3 increased from 460.6

kN to 842.2 kN after the failure of the blade pitch control. This change is very critical for

the structural integrity of the tower, and it may cause the collapse of the turbine tower.

Tower base torsional moment in Figure 8.26 shows that the changes after failure

did not make a significant difference. The maximum moment of turbine #3 increased by

13.5% after the failure event; it was seen that this amount of change was relatively small

compared to the responses presented earlier.

Figure 8.27 Top view of mooring-line arrangement (Rectangular platform)

When turbine #3 fails, the mooring line top tensions are also affected by that

emergency. Since the blade pitch angle of turbine #3 is assumed to be decreased, the

total aero dynamic force on this turbine is increased, and the platform yaw moment is

induced. As a result, the mean top tensions of lines #1 and #3 in Figure 8.27 are

Line #1

Line #2

Line #3

WINDWAVE

Line #4

192

noticeably increased while the top tension of line #2 is reduced because of the shortened

length between the fairlead and anchor position. The maximum top tension of line #1

increased from 1,299 kN to 1,410 kN and that of line #3 increased from 3,008 kN to

3,361 kN after the failure event. On the contrary, the maximum top tension of line #2

decreased from 1,351 kN to 1,234 kN. The time series and its spectra are presented in

Figure 8.28.

In the case of line #4, the mean top tension slightly increased after the failure

event due to the combination of larger surge drift and backward motion due to the

negative platform yaw. Thus, the variations of mooring line top tensions in an

emergency situation are very difficult to estimate without this numerical analysis tool for

MUFOWT.

Lin

e #

1 L

ine

#2

(a) (b) Figure 8.28 Top-tension (a) and spectra (b) with blade control failure


0 200 400 600 800 10001000

1200

1400

1600

1800

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1

2

3x 10

4

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001000

1200

1400

1600

1800

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

2

4

6

8x 10

4

(rad/sec)

S(

)

Normal

Fault

193

Lin

e #

3 L

ine

#4


8.6 Discussion

So far, the coupled dynamic analysis of the five-turbine MUFOWT has been

designed and analyzed in this chapter. This semi-submersible platform was equipped

with five 5MW baseline turbines so the total rated power of this unit is equivalent to

25MW. The hydrostatic equilibrium was carefully checked by rigorous calculations of

the mass properties of turbine and platform. The natural frequencies of the 6-DOFs of

platform mode were obtained from the free decay test which revealed that the platform is

well designed for the wind, wave environment. In normal operational conditions, the

platform and turbine responses seemed to be stable and the power production of all

turbines looked normal. The overall performance of MUFOWT with one turbine failure

event is also investigated. If one of the turbines does not work normally, then the overall

aerodynamic equilibrium cannot be guaranteed. As a result, unexpected platform yaw

0 200 400 600 800 10001000

2000

3000

4000

5000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

5

10

15x 10

5

(rad/sec)

S(

)

Normal

Fault

0 200 400 600 800 10001000

2000

3000

4000

5000

Time(sec)

kN

Normal

Fault

0 0.5 1 1.5 20

1

2

3

4x 10

5

(rad/sec)

S(

)

Normal

Fault

194

moment is induced, and the turbine local structural loads can either be increased or

decreased depending on the failure event. This failure also affects the mooring line top

tensions. Depending on the mooring line arrangement and the direction of platform

offset, the mooring line can be more taut or slackened as presented in this study.

195

9. CONCLUSION AND FUTURE WORK

9.1 Coupled Dynamic Analysis of MUFOWT

So far, most offshore wind turbine research has been limited to fixed towers in

shallow-water areas. This study investigated a floating offshore wind turbine that can be

used in deeper waters. The numerical tool combined with CHARM3D was able to

analyze rotor-floater-tether coupled nonlinear dynamics in the time domain and was used

for floater-motion, tower-acceleration and mooring-line-tension simulations in a

collinear wind-wave environment. The coupled analysis has included time-varying

aerodynamic loading, tower-blade elastic deformation, blade-control-induced loading,

and gyroscopic effect in analyzing the global dynamics of the entire system. As a simple

case study, the rotor-floater coupling effects of the Hywind spar were assessed through

comparisons with the results of an uncoupled analysis in which the whole body was

treated as a rigid body. The influence of control strategies were also simulated and

reviewed.

The development of a numerical analysis tool for MUFOWT was one of the most

challenging aspects of this study. One of the most popular wind turbine simulation tools

‘FAST’ was limited to a land-based turbine or single-turbine floating platform. Even so,

various floating platforms including TLP, Spar, barge and semi-submersible were tested

and performances were evaluated with this tool. To analyze multiple turbines on one

floating platform, significant modification of current tools was required and was

successfully done throughout this study.

196

The development of new analysis tools for MUFOWT started from the single

turbine analysis tool FAST. It was then expanded to include a number of turbine

variables and functions. In addition, the aerodynamic calculation module should also be

expanded to calculate the aerodynamic loadings on different turbines. Above all, the

design of the global matrix and forcing function of MUFOWT was the most important

work. Based on the formation of a single turbine matrix, the global matrix for one

floating platform with multiple turbines was suggested and implemented in this research.

Similarly, the forcing function for MUFOWT was also designed and included. To

evaluate this developed tool, two semi-submersible types of floating platforms with three

and five turbines were proposed and analyzed. System identification work with a free

decay test was conducted in advance followed by a time marching simulation with the

wind, wave environment. All the dynamic aspects of multiple turbines with a floating

platform were effectively captured with this numerical tool and the design of a semi-

submersible platform was also validated.

9.2 Future Work

One of the important limitations of this study is the consideration of interference

effects between adjacent turbines. Numerically, those effects cannot be included at this

moment because the wake and turbulence wind field made by the rotor blade and the

influence on the other rotating blade is difficult to measure with the current technology.

Instead, the location of multiple turbines on one floating platform was carefully selected

so that the shade effect was minimized. In the future, when more advanced aerodynamic

197

computations are available including the shade and interference effect, this tool can more

accurately treat the dynamic aspects of MUFOWT. Furthermore, the floating platform

can be also modeled as an elastic body. The current platform has only 6 degrees of

freedom for translational and rotational motion, but as the platform gets bigger, the

elastic modes of the platform should also be considered. Eventually, the fully coupled

turbine-elastic platform-mooring can be utilized for the analysis of MUFOWT.

The development of remote structural health monitoring systems for MUFOWT

can be also promoted using currently developed tools because the various scenarios of

floating platform and multiple-turbine responses can be utilized as a database system

using the simulation results. For example, the abnormal signal from the unmanned

MUFOWT system can be detected by a monitoring system and a remote operator can

analyze the specific problems of the turbine system using the generated database.

198

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